Assertion-Reason MCQs for Sub-Topics of Topic 12: Vectors & Three-Dimensional Geometry

Question 1.

Assertion (A): The sum of two vectors $\vec{a}$ and $\vec{b}$ is always a vector quantity.

Reason (R): Vector addition is defined such that the resultant has both magnitude and direction.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2.

Assertion (A): The zero vector has zero magnitude and its direction is not uniquely defined.

Reason (R): Division by zero is undefined, so a direction cannot be assigned to a vector with zero magnitude in the usual sense.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3.

Assertion (A): Two vectors $\vec{a}$ and $\vec{b}$ are equal if and only if $|\vec{a}| = |\vec{b}|$.

Reason (R): Equal vectors must have both the same magnitude and the same direction.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4.

Assertion (A): For any scalar $k$ and vector $\vec{a}$, the magnitude of $k\vec{a}$ is $|k||\vec{a}|$.

Reason (R): Scalar multiplication scales the length of the vector by the absolute value of the scalar, and preserves or reverses the direction depending on the sign of the scalar.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5.

Assertion (A): Vector addition is commutative, i.e., $\vec{a} + \vec{b} = \vec{b} + \vec{a}$.

Reason (R): This property is demonstrated by the parallelogram law of vector addition, where the diagonal represents the sum regardless of which vector is added first.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 6.

Assertion (A): If two vectors $\vec{a}$ and $\vec{b}$ are collinear, then $\vec{a} = k\vec{b}$ for some scalar $k$.

Reason (R): Collinear vectors lie on the same line or parallel lines, meaning their directions are either the same or opposite, which is captured by scalar multiplication.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1.

Assertion (A): The position vector of a point $(x, y, z)$ in 3D space is $x\hat{i} + y\hat{j} + z\hat{k}$.

Reason (R): A position vector originates from the origin and terminates at the given point, and $\hat{i}, \hat{j}, \hat{k}$ are unit vectors along the axes.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2.

Assertion (A): The vector joining point A with position vector $\vec{a}$ to point B with position vector $\vec{b}$ is $\vec{b} - \vec{a}$.

Reason (R): $\vec{AB} = \vec{OB} - \vec{OA} = \vec{b} - \vec{a}$ according to the triangle law of vector addition where $\vec{OA} + \vec{AB} = \vec{OB}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3.

Assertion (A): The magnitude of the vector $\vec{v} = 2\hat{i} - \hat{j} + 2\hat{k}$ is 3.

Reason (R): The magnitude of a vector $x\hat{i} + y\hat{j} + z\hat{k}$ is $\sqrt{x^2+y^2+z^2}$. For $\vec{v}$, $|\vec{v}| = \sqrt{2^2 + (-1)^2 + 2^2} = \sqrt{4+1+4} = \sqrt{9} = 3$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4.

Assertion (A): Any vector in a 2D plane can be expressed as a linear combination of $\hat{i}$ and $\hat{j}$.

Reason (R): $\hat{i}$ and $\hat{j}$ are unit vectors along the perpendicular coordinate axes and form a basis for the 2D plane.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5.

Assertion (A): If vectors $\vec{a}, \vec{b}, \vec{c}$ are coplanar and $\vec{a}, \vec{b}$ are non-collinear, then $\vec{c}$ can always be written as a linear combination of $\vec{a}$ and $\vec{b}$.

Reason (R): Coplanar vectors lie in the same plane, and if two vectors in that plane are non-collinear, they form a basis for that plane.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 6.

Assertion (A): If a vector $\vec{v} = x\hat{i} + y\hat{j} + z\hat{k}$ is a unit vector, then $x^2+y^2+z^2 = 1$.

Reason (R): A vector $\vec{v} = x\hat{i} + y\hat{j} + z\hat{k}$ is a unit vector if its magnitude $\sqrt{x^2+y^2+z^2} = 1$, which implies $x^2+y^2+z^2 = 1$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1.

Assertion (A): The dot product of two perpendicular vectors is always zero.

Reason (R): The formula for the dot product is $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$, and $\cos 90^\circ = 0$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2.

Assertion (A): The work done by a force is a scalar quantity.

Reason (R): Work done is calculated as the dot product of the force vector and the displacement vector.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3.

Assertion (A): For any vector $\vec{a}$, $\vec{a} \cdot \vec{a} = |\vec{a}|^2$.

Reason (R): The angle between a vector and itself is $0^\circ$, and $\cos 0^\circ = 1$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4.

Assertion (A): The projection of vector $\vec{a}$ on vector $\vec{b}$ is $\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|}$.

Reason (R): The scalar projection of $\vec{a}$ on $\vec{b}$ is given by $|\vec{a}|\cos\theta = \frac{|\vec{a}||\vec{b}|\cos\theta}{|\vec{b}|} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5.

Assertion (A): If $\vec{a} \cdot \vec{b} > 0$, the angle between non-zero vectors $\vec{a}$ and $\vec{b}$ is acute.

Reason (R): $\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$. If $\vec{a} \cdot \vec{b} > 0$ and magnitudes are positive, then $\cos\theta > 0$. The angle $\theta$ between vectors is in $[0, \pi]$, and $\cos\theta > 0$ in $[0, \pi/2)$, which is acute.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1.

Assertion (A): The cross product of two parallel non-zero vectors is the zero vector.

Reason (R): The magnitude of the cross product is $|\vec{a}||\vec{b}|\sin\theta$, and for parallel vectors $\theta = 0^\circ$ or $180^\circ$, so $\sin\theta = 0$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2.

Assertion (A): $\vec{a} \times \vec{b} = \vec{b} \times \vec{a}$ for any two vectors $\vec{a}$ and $\vec{b}$.

Reason (R): The cross product is anti-commutative, i.e., $\vec{a} \times \vec{b} = -\vec{b} \times \vec{a}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3.

Assertion (A): The magnitude of the cross product of two vectors represents the area of the parallelogram formed by them as adjacent sides.

Reason (R): The area of a parallelogram with sides $|\vec{a}|$ and $|\vec{b}|$ and angle $\theta$ between them is $|\vec{a}||\vec{b}|\sin\theta$, which is equal to $|\vec{a} \times \vec{b}|$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4.

Assertion (A): The vector $\vec{a} \times \vec{b}$ is perpendicular to the plane containing $\vec{a}$ and $\vec{b}$.

Reason (R): The direction of the cross product is defined using the right-hand rule such that it is orthogonal to both vectors.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5.

Assertion (A): If $\vec{a} = 2\hat{i} - \hat{j}$ and $\vec{b} = 4\hat{i} - 2\hat{j}$, then $\vec{a} \times \vec{b} = \vec{0}$.

Reason (R): $\vec{b} = 2\vec{a}$, so the vectors $\vec{a}$ and $\vec{b}$ are parallel, and the cross product of parallel vectors is the zero vector.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 6.

Assertion (A): The area of a triangle with vertices A, B, C is $\frac{1}{2} |\vec{AB} \times \vec{AC}|$.

Reason (R): The area of a triangle is half the area of the parallelogram formed by two of its sides as adjacent vectors, and $|\vec{AB} \times \vec{AC}|$ is the area of the parallelogram formed by $\vec{AB}$ and $\vec{AC}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1.

Assertion (A): The scalar triple product $[\vec{a}, \vec{b}, \vec{c}]$ is a scalar quantity.

Reason (R): It is calculated as the dot product of one vector with the cross product of the other two, and the dot product of two vectors is a scalar.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2.

Assertion (A): If $[\vec{a}, \vec{b}, \vec{c}] = 0$, then the vectors $\vec{a}, \vec{b}, \vec{c}$ are coplanar.

Reason (R): The absolute value of the scalar triple product represents the volume of the parallelepiped formed by the three vectors, and if this volume is zero, the vectors must lie in the same plane.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3.

Assertion (A): $[\vec{a}, \vec{b}, \vec{c}] = [\vec{b}, \vec{c}, \vec{a}]$.

Reason (R): Cyclical permutation of the vectors in the scalar triple product does not change its value.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4.

Assertion (A): The volume of the tetrahedron formed by coterminous edges $\vec{a}, \vec{b}, \vec{c}$ is $\frac{1}{6}|[\vec{a}, \vec{b}, \vec{c}]|$.

Reason (R): The volume of the parallelepiped is $|[\vec{a}, \vec{b}, \vec{c}]|$, and the volume of the tetrahedron sharing the same coterminous edges is one-sixth of the volume of the parallelepiped.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5.

Assertion (A): If any two vectors in a scalar triple product are parallel, the value of the product is zero.

Reason (R): If $\vec{a} \parallel \vec{b}$, then $\vec{a} \times \vec{b} = \vec{0}$, and the scalar triple product $[\vec{a}, \vec{b}, \vec{c}] = (\vec{a} \times \vec{b}) \cdot \vec{c} = \vec{0} \cdot \vec{c} = 0$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 6.

Assertion (A): The vectors $\hat{i}, \hat{j}, \hat{k}$ are non-coplanar.

Reason (R): Their scalar triple product $[\hat{i}, \hat{j}, \hat{k}] = 1 \neq 0$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1.

Assertion (A): The position vector of the midpoint of the line segment joining $\vec{a}$ and $\vec{b}$ is $\frac{\vec{a} + \vec{b}}{2}$.

Reason (R): The midpoint divides the line segment internally in the ratio 1:1, and the section formula for internal division is $\frac{n\vec{a} + m\vec{b}}{m+n}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2.

Assertion (A): If point R divides the line segment PQ internally in the ratio $m:n$, then $m+n$ is the sum of the ratio terms.

Reason (R): In the internal section formula $\vec{r} = \frac{n\vec{p} + m\vec{q}}{m+n}$, the denominator is the sum of the ratio terms used in the numerator.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3.

Assertion (A): The position vector of the centroid of a triangle with vertices $\vec{a}, \vec{b}, \vec{c}$ is the average of the position vectors, i.e., $\frac{\vec{a}+\vec{b}+\vec{c}}{3}$.

Reason (R): The centroid is the point of intersection of the medians, and it divides each median in the ratio 2:1.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4.

Assertion (A): The formula for external division of a line segment AB in the ratio $m:n$ gives the position vector of a point R such that R lies on the line AB but outside the segment AB (unless $m=n$).

Reason (R): The section formula for external division is $\vec{r} = \frac{m\vec{b} - n\vec{a}}{m-n}$, which corresponds to the ratio $m:n$ with one of the ratio terms being effectively negative for internal division.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5.

Assertion (A): If the position vectors of three points A, B, C are $\vec{a}, \vec{b}, \vec{c}$ and $\vec{c}$ can be expressed as $\vec{c} = (1-t)\vec{a} + t\vec{b}$ for some scalar $t$, then A, B, C are collinear.

Reason (R): The expression $\vec{r} = (1-\lambda)\vec{a} + \lambda\vec{b}$ is the vector equation of the line passing through points with position vectors $\vec{a}$ and $\vec{b}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 6.

Assertion (A): The position vector of the centroid of a tetrahedron is the average of the position vectors of its four vertices.

Reason (R): The centroid of a tetrahedron divides the line joining any vertex to the centroid of the opposite face in the ratio 3:1.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1.

Assertion (A): The equation of the yz-plane in 3D space is $x=0$.

Reason (R): Any point on the yz-plane has its x-coordinate equal to zero, while y and z can take any real values.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2.

Assertion (A): The direction ratios of the line joining points P$(1,2,3)$ and Q$(4,5,6)$ are $(3,3,3)$.

Reason (R): The direction ratios of the line joining $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ are proportional to $(x_2-x_1, y_2-y_1, z_2-z_1)$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3.

Assertion (A): The vector equation of a line passing through point $\vec{a}$ and parallel to vector $\vec{b}$ is $\vec{r} = \vec{a} + \lambda \vec{b}$.

Reason (R): Any point on the line, relative to the origin ($\vec{r}$), is the sum of a fixed point on the line ($\vec{a}$) and a variable displacement from that point along the direction of the line ($\lambda \vec{b}$).

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4.

Assertion (A): The sum of the squares of the direction cosines of any line is always 1.

Reason (R): Direction cosines are the components of a unit vector along the line.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5.

Assertion (A): The Cartesian equation of the line $\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda (4\hat{i} + 5\hat{j} + 6\hat{k})$ is $\frac{x-1}{4} = \frac{y-2}{5} = \frac{z-3}{6}$.

Reason (R): If a line passes through $(x_1, y_1, z_1)$ and has direction ratios $(a, b, c)$, its Cartesian equation is $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 6.

Assertion (A): The direction cosines of the line $\vec{r} = \hat{i} + \lambda \hat{i}$ are $(1,0,0)$.

Reason (R): The direction vector is $\hat{i}$, which is along the x-axis. The direction cosines of the x-axis are $(1,0,0)$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1.

Assertion (A): The vector $\vec{n} = A\hat{i} + B\hat{j} + C\hat{k}$ is a normal vector to the plane $Ax + By + Cz + D = 0$.

Reason (R): For any two points P and Q on the plane, the vector $\vec{PQ}$ lies in the plane, and its dot product with the normal vector is zero.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2.

Assertion (A): The distance of the plane $2x - y + 2z = 6$ from the origin is 2.

Reason (R): The distance of the plane $Ax + By + Cz + D = 0$ from the origin is $\frac{|D|}{\sqrt{A^2+B^2+C^2}}$. In this case, $A=2, B=-1, C=2, D=-6$, so distance is $\frac{|-6|}{\sqrt{2^2+(-1)^2+2^2}} = \frac{6}{\sqrt{4+1+4}} = \frac{6}{3} = 2$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3.

Assertion (A): The equation of a plane passing through three collinear points is unique.

Reason (R): A plane is uniquely determined by three non-collinear points.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4.

Assertion (A): The equation $x+y+z=1$ represents a plane making equal intercepts on the coordinate axes.

Reason (R): The intercept form of the plane equation is $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$, where $a, b, c$ are the intercepts on the x, y, z axes respectively. For $x+y+z=1$, $a=b=c=1$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5.

Assertion (A): The equation $\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 0$ represents a plane passing through the origin.

Reason (R): If a plane passes through the origin, the perpendicular distance from the origin to the plane is zero, so the constant term $d$ in $\vec{r} \cdot \hat{n} = d$ (or $D$ in $Ax+By+Cz+D=0$) is zero.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 6.

Assertion (A): The plane $2x - y + 3z = 5$ and the plane $4x - 2y + 6z = 10$ are identical.

Reason (R): Two planes $A_1x + B_1y + C_1z + D_1 = 0$ and $A_2x + B_2y + C_2z + D_2 = 0$ are identical if $\frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2} = \frac{D_1}{D_2}$. Here $\frac{2}{4} = \frac{-1}{-2} = \frac{3}{6} = \frac{-5}{-10} = \frac{1}{2}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1.

Assertion (A): If two lines have direction ratios $(1, 2, 3)$ and $(-2, -4, -6)$, the angle between them is $0^\circ$.

Reason (R): The direction ratios are proportional ($\frac{1}{-2} = \frac{2}{-4} = \frac{3}{-6} = -\frac{1}{2}$), indicating the lines are parallel or collinear. Since $k$ is negative, they are parallel and point in opposite directions, making the angle $180^\circ$ or $\pi$ radians.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2.

Assertion (A): The angle between the lines $\vec{r} = \vec{a}_1 + \lambda \vec{b}_1$ and $\vec{r} = \vec{a}_2 + \mu \vec{b}_2$ is the angle between $\vec{b}_1$ and $\vec{b}_2$.

Reason (R): The vectors $\vec{b}_1$ and $\vec{b}_2$ represent the directions of the lines.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3.

Assertion (A): Two planes $A_1x + B_1y + C_1z + D_1 = 0$ and $A_2x + B_2y + C_2z + D_2 = 0$ are parallel if $A_1A_2 + B_1B_2 + C_1C_2 = 0$.

Reason (R): The condition for parallel planes is that their normal vectors are parallel, which means their components are proportional, i.e., $\frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2}$. The dot product condition $A_1A_2 + B_1B_2 + C_1C_2 = 0$ is for perpendicular planes.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4.

Assertion (A): The angle between a line and a plane is the complement of the angle between the line and the normal to the plane.

Reason (R): If $\theta$ is the angle between the line and the plane's normal, the angle between the line and the plane itself is $\phi = 90^\circ - \theta$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5.

Assertion (A): The line $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ is parallel to the plane $2x + 3y + 4z = 0$.

Reason (R): A line with direction vector $\vec{b}$ is parallel to a plane with normal vector $\vec{n}$ if $\vec{b} \cdot \vec{n} = 0$. Here, $\vec{b} = 2\hat{i} + 3\hat{j} + 4\hat{k}$ and $\vec{n} = 2\hat{i} + 3\hat{j} + 4\hat{k}$. $\vec{b} \cdot \vec{n} = (2)(2) + (3)(3) + (4)(4) = 4 + 9 + 16 = 29 \neq 0$. The line is not parallel to the plane; it is perpendicular to it.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 6.

Assertion (A): The angle between the plane $z=0$ and the plane $x=0$ is $90^\circ$.

Reason (R): The normal vector to the plane $z=0$ is $\hat{k}$ (or $(0,0,1)$) and the normal vector to the plane $x=0$ is $\hat{i}$ (or $(1,0,0)$). The dot product of these normal vectors is $\hat{k} \cdot \hat{i} = 0$, which implies the planes are perpendicular.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1.

Assertion (A): The shortest distance between two intersecting lines in 3D space is zero.

Reason (R): Intersecting lines share a common point, so the minimum distance between any point on one line and any point on the other line is the distance between the point of intersection and itself, which is zero.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2.

Assertion (A): The shortest distance between two parallel lines $\vec{r} = \vec{a}_1 + \lambda \vec{b}$ and $\vec{r} = \vec{a}_2 + \mu \vec{b}$ is given by $\frac{|(\vec{a}_2 - \vec{a}_1) \times \vec{b}|}{|\vec{b}|}$.

Reason (R): $(\vec{a}_2 - \vec{a}_1) \times \vec{b}$ represents the vector area of the parallelogram formed by the vector joining points on the lines and the common direction vector, and dividing by the base $|\vec{b}|$ gives the height, which is the shortest distance.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3.

Assertion (A): The distance of the point $(1, 2, 3)$ from the plane $x+y+z = 6$ is 0.

Reason (R): Substituting the point $(1,2,3)$ into the equation of the plane gives $1+2+3=6$, which is true. This means the point lies on the plane, and the distance of a point from a plane it lies on is zero.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4.

Assertion (A): Skew lines are non-parallel lines that intersect.

Reason (R): Skew lines are lines in 3D space that are neither parallel nor intersecting. They exist in different planes.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5.

Assertion (A): The shortest distance between the parallel planes $2x-y+z=1$ and $2x-y+z=5$ is $\frac{4}{\sqrt{6}}$.

Reason (R): The distance between two parallel planes $Ax+By+Cz+D_1=0$ and $Ax+By+Cz+D_2=0$ is $\frac{|D_2-D_1|}{\sqrt{A^2+B^2+C^2}}$. Here $A=2, B=-1, C=1, D_1=-1, D_2=-5$. Distance = $\frac{|-5 - (-1)|}{\sqrt{2^2+(-1)^2+1^2}} = \frac{|-4|}{\sqrt{4+1+1}} = \frac{4}{\sqrt{6}}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 6.

Assertion (A): The shortest distance between two skew lines is measured along a line segment perpendicular to both lines.

Reason (R): This unique line segment is the common perpendicular, and its length is the minimum distance between the two skew lines.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.