Matching Items MCQs for Sub-Topics of Topic 12: Vectors & Three-Dimensional Geometry

(i) Unit Vector

(ii) Zero Vector

(iii) Collinear Vectors

(iv) Coinitial Vectors

(v) Equal Vectors

(a) Have the same initial point.

(b) Magnitude is 1.

(c) Lie on the same line or parallel lines.

(d) Magnitude is 0.

(e) Have same magnitude and direction.

(i) Commutativity of Addition

(ii) Associativity of Addition

(iii) Identity for Addition

(iv) Distributivity of Scalar Multiplication over Vector Addition

(v) Associativity of Scalar Multiplication

(a) $k_1(k_2\vec{a}) = (k_1k_2)\vec{a}$

(b) $\vec{a} + \vec{b} = \vec{b} + \vec{a}$

(c) $\vec{a} + \vec{0} = \vec{a}$

(d) $k(\vec{a} + \vec{b}) = k\vec{a} + k\vec{b}$

(e) $(\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})$

(i) Speed

(ii) Acceleration

(iii) Distance

(iv) Force

(v) Mass

(a) Vector

(b) Scalar

(i) $\vec{a}$

(ii) $|\vec{a}|$

(iii) $-\vec{a}$

(iv) $\hat{a}$

(v) $\vec{AB}$

(a) Magnitude of vector $\vec{a}$.

(b) A vector quantity.

(c) A vector from point A to point B.

(d) A unit vector in the direction of $\vec{a}$.

(e) A vector with the same magnitude as $\vec{a}$ but opposite direction.

(i) $\vec{v} + \vec{0}$

(ii) $\vec{v} - \vec{v}$

(iii) $k \cdot \vec{0}$ (scalar $k$)

(iv) $1 \cdot \vec{v}$

(v) $0 \cdot \vec{v}$

(a) $\vec{0}$

(b) $\vec{v}$

(i) Position vector of (2, -1, 3)

(ii) Vector from (1, 2) to (4, 6)

(iii) Unit vector along positive z-axis

(iv) Vector with magnitude 5 along positive x-axis

(v) Zero vector in 3D

(a) $5\hat{i}$

(b) $3\hat{i} + 4\hat{j}$

(c) $2\hat{i} - \hat{j} + 3\hat{k}$

(d) $\vec{0}$

(e) $\hat{k}$

(i) $\vec{v} = x\hat{i} + y\hat{j}$ is a unit vector

(ii) Vectors $\vec{a}=(a_1,a_2,a_3)$ and $\vec{b}=(b_1,b_2,b_3)$ are collinear

(iii) Position vector of origin

(iv) Magnitude of $\vec{v} = x\hat{i} + y\hat{j} + z\hat{k}$

(v) Collinearity of points A, B, C with position vectors $\vec{a}, \vec{b}, \vec{c}$

(a) $\vec{0}$

(b) $\vec{AB} = k \vec{AC}$ for some scalar $k$

(c) $\frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3}$ (if denominators are non-zero)

(d) $x^2 + y^2 = 1$

(e) $\sqrt{x^2 + y^2 + z^2}$

(i) Linear Combination

(ii) Linearly Dependent Vectors

(iii) Basis Vectors in 2D

(iv) Coplanar Vectors (non-zero)

(v) Standard Basis in 3D

(a) $\hat{i}, \hat{j}, \hat{k}$

(b) One vector is a linear combination of the others.

(c) An expression of the form $c_1\vec{v}_1 + c_2\vec{v}_2 + \dots + c_n\vec{v}_n$

(d) Any three vectors whose scalar triple product is zero.

(e) Two non-zero, non-collinear vectors in a plane.

(i) $(a_1\hat{i} + a_2\hat{j}) + (b_1\hat{i} + b_2\hat{j})$

(ii) $(a_1\hat{i} + a_2\hat{j} + a_3\hat{k}) - (b_1\hat{i} + b_2\hat{j} + b_3\hat{k})$

(iii) $k(a_1\hat{i} + a_2\hat{j} + a_3\hat{k})$

(iv) $\frac{1}{|\vec{v}|}\vec{v}$ where $\vec{v} = x\hat{i} + y\hat{j} + z\hat{k}$

(v) Vector joining P$(x_1, y_1)$ to Q$(x_2, y_2)$

(a) $(x_2-x_1)\hat{i} + (y_2-y_1)\hat{j}$

(b) $(a_1+b_1)\hat{i} + (a_2+b_2)\hat{j}$

(c) $\frac{x\hat{i} + y\hat{j} + z\hat{k}}{\sqrt{x^2+y^2+z^2}}$

(d) $ka_1\hat{i} + ka_2\hat{j} + ka_3\hat{k}$

(e) $(a_1-b_1)\hat{i} + (a_2-b_2)\hat{j} + (a_3-b_3)\hat{k}$

(i) Vector parallel to y-axis

(ii) Vector in xz-plane

(iii) Vector with zero magnitude

(iv) Vector $\vec{v}=v_x\hat{i} + v_y\hat{j}$ scaled by 2

(v) Vector $\vec{AB}$ where A and B have same x-coordinate

(a) $v_x=0, v_z=0$

(b) $v_x=0$ for $\vec{AB}$

(c) $v_y=0$

(d) $v_x=0, v_y=0, v_z=0$

(e) $2v_x\hat{i} + 2v_y\hat{j}$

(i) $\hat{i} \cdot \hat{j}$

(ii) $\hat{k} \cdot \hat{k}$

(iii) $\vec{a} \cdot \vec{b}$ if $\vec{a} \perp \vec{b}$

(iv) $\vec{a} \cdot \vec{0}$

(v) $\vec{a} \cdot \vec{a}$

(a) $|\vec{a}|^2$

(b) 0

(c) 1

(i) Work Done

(ii) Angle between vectors $\vec{a}, \vec{b}$

(iii) Scalar projection of $\vec{a}$ on $\vec{b}$

(iv) Vector projection of $\vec{a}$ on $\vec{b}$

(v) $\vec{a} \cdot \vec{b}$ in components

(a) $\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$

(b) $\vec{F} \cdot \vec{d}$

(c) $a_1b_1 + a_2b_2 + a_3b_3$

(d) $\cos^{-1}\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}\right)$

(e) $(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2})\vec{b}$

(i) $\vec{a} \cdot \vec{b} > 0$

(ii) $\vec{a} \cdot \vec{b} < 0$

(iii) $\vec{a} \cdot \vec{b} = 0$

(iv) $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|$

(v) $\vec{a} \cdot \vec{b} = -|\vec{a}||\vec{b}|$

(a) Angle is $180^\circ$ (parallel in opposite direction).

(b) Angle is $90^\circ$ (perpendicular).

(c) Angle is acute.

(d) Angle is $0^\circ$ (parallel in same direction).

(e) Angle is obtuse.

(i) $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$

(ii) $\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$

(iii) $(k\vec{a}) \cdot \vec{b} = k(\vec{a} \cdot \vec{b})$

(iv) $\vec{a} \cdot \vec{a} \ge 0$

(v) $\vec{a} \cdot \vec{0} = 0$

(a) Property of non-negativity

(b) Property of zero vector

(c) Commutativity

(d) Associativity with scalar

(e) Distributivity

(i) $\hat{i} + \hat{j}$ and $\hat{i} - \hat{j}$

(ii) $2\hat{i}$ and $3\hat{i}$

(iii) $\hat{i} + 2\hat{j} + 3\hat{k}$ and $2\hat{i} + \hat{j} + \hat{k}$

(iv) $\hat{i} + \hat{j} + \hat{k}$ and $\hat{i} + \hat{j} + \hat{k}$

(v) $2\hat{i} + \hat{j}$ and $\hat{i} - 2\hat{j}$

(a) 6

(b) 0

(c) 3

(d) 5

(i) $\hat{i} \times \hat{j}$

(ii) $\hat{k} \times \hat{k}$

(iii) $\vec{a} \times \vec{b}$ if $\vec{a} \parallel \vec{b}$

(iv) $\vec{a} \times \vec{0}$

(v) $\vec{a} \times \vec{a}$

(a) $\vec{0}$

(b) $\hat{k}$

(i) Area of parallelogram with sides $\vec{a}, \vec{b}$

(ii) Area of triangle with sides $\vec{a}, \vec{b}$

(iii) A vector perpendicular to $\vec{a}$ and $\vec{b}$

(iv) Torque $\vec{\tau}$ by force $\vec{F}$ at position $\vec{r}$

(v) Unit vector perpendicular to $\vec{a}$ and $\vec{b}$

(a) $\vec{a} \times \vec{b}$

(b) $\frac{1}{2}|\vec{a} \times \vec{b}|$

(c) $|\vec{a} \times \vec{b}|$

(d) $\vec{r} \times \vec{F}$

(e) $\frac{\vec{a} \times \vec{b}}{|\vec{a} \times \vec{b}|}$

(i) $|\vec{a} \times \vec{b}| > 0$

(ii) $|\vec{a} \times \vec{b}| = 0$

(iii) $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|$

(iv) $\vec{a} \times \vec{b}$ direction is along $\hat{n}$

(v) $\vec{b} \times \vec{a}$ direction is along $-\hat{n}$

(a) Vectors are parallel/collinear.

(b) Right-hand rule.

(c) Vectors are perpendicular.

(d) Vectors are not parallel/collinear.

(e) Anti-commutativity.

(i) $\vec{a} \times \vec{b} = -\vec{b} \times \vec{a}$

(ii) $\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}$

(iii) $(k\vec{a}) \times \vec{b} = k(\vec{a} \times \vec{b})$

(iv) $\vec{a} \times \vec{a} = \vec{0}$

(v) $\vec{a} \times (\vec{b} \times \vec{c})$

(a) Vector triple product.

(b) Property of parallel vectors.

(c) Distributivity over vector addition.

(d) Associativity with scalar.

(e) Anti-commutativity.

(i) $\hat{i} + \hat{j}$ and $\hat{i} - \hat{j}$

(ii) $2\hat{i}$ and $3\hat{i}$

(iii) $\hat{i} + \hat{j}$ and $\hat{j}$

(iv) $2\hat{i}$ and $3\hat{j}$

(v) $\hat{i}$ and $\hat{j}$

(a) 0

(b) 1

(c) 2

(d) 6

(i) $[\vec{a}, \vec{b}, \vec{c}]$

(ii) $[\vec{a}, \vec{b}, \vec{c}]$ using components

(iii) $[\vec{a}, \vec{a}, \vec{c}]$

(iv) $[\hat{i}, \hat{j}, \hat{k}]$

(v) $[\vec{a}, \vec{b}, \vec{c}]$ vs $[\vec{b}, \vec{c}, \vec{a}]$

(a) Value is 0.

(b) $\vec{a} \cdot (\vec{b} \times \vec{c})$

(c) Value is 1.

(d) Value is the same.

(e) Determinant $\begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}$.

(i) $|[\vec{a}, \vec{b}, \vec{c}]|$

(ii) $\frac{1}{6}|[\vec{a}, \vec{b}, \vec{c}]|$

(iii) $[\vec{a}, \vec{b}, \vec{c}] = 0$

(iv) $[\vec{a}, \vec{b}, \vec{c}] \neq 0$

(v) $[\vec{a}, \vec{b}, \vec{c}] > 0$

(a) Vectors are coplanar.

(b) Volume of parallelepiped.

(c) Vectors are non-coplanar.

(d) Vectors form a right-handed system.

(e) Volume of tetrahedron.

(i) Permutation

(ii) Identical Vectors

(iii) Scalar Multiplication

(iv) Distributivity

(v) Dot and Cross Interchange

(a) $[\vec{a}+\vec{d}, \vec{b}, \vec{c}] = [\vec{a}, \vec{b}, \vec{c}] + [\vec{d}, \vec{b}, \vec{c}]$

(b) $\vec{a} \cdot (\vec{b} \times \vec{c}) = (\vec{a} \times \vec{b}) \cdot \vec{c}$

(c) $[\vec{a}, \vec{a}, \vec{b}] = 0$

(d) $[k\vec{a}, \vec{b}, \vec{c}] = k[\vec{a}, \vec{b}, \vec{c}]$

(e) $[\vec{a}, \vec{b}, \vec{c}] = [\vec{b}, \vec{c}, \vec{a}]$

(i) $\{\hat{i}, \hat{j}, \hat{k}\}$

(ii) $\{\hat{i} + \hat{j}, \hat{j} + \hat{k}, \hat{i} + \hat{k}\}$

(iii) $\{\hat{i} - \hat{j}, \hat{j} - \hat{k}, \hat{k} - \hat{i}\}$

(iv) $\{2\hat{i}, 3\hat{j}, 4\hat{k}\}$

(v) $\{\vec{a}, \vec{b}, \vec{a}+\vec{b}\}$

(a) 24

(b) 2

(c) 0 (always coplanar)

(d) 0 (always coplanar)

(e) 1

(i) Condition for coplanarity of $\vec{a}, \vec{b}, \vec{c}$

(ii) Volume of parallelepiped is zero

(iii) Vectors are linearly dependent (non-trivial linear combination is zero)

(iv) Points A, B, C, D are coplanar

(v) $\vec{a} \cdot (\vec{b} \times \vec{c})$

(a) Same as $[\vec{a}, \vec{b}, \vec{c}]$

(b) $[\vec{a}, \vec{b}, \vec{c}] = 0$

(c) $[\vec{AB}, \vec{AC}, \vec{AD}] = 0$

(d) $[\vec{a}, \vec{b}, \vec{c}] = 0$

(e) $[\vec{a}, \vec{b}, \vec{c}] = 0$

(i) Internal Division (R on AB)

(ii) External Division (R on AB extended, beyond B, $m>n$)

(iii) External Division (R on AB extended, beyond A, $m

(iv) Midpoint of AB

(v) Point dividing AB in ratio 1:2 internally

(a) $\frac{\vec{a} + 2\vec{b}}{3}$

(b) $\frac{m\vec{b} - n\vec{a}}{m-n}$

(c) $\frac{n\vec{a} + m\vec{b}}{m+n}$

(d) $\frac{m\vec{b} - n\vec{a}}{m-n}$ (where $m-n$ is negative)

(e) $\frac{\vec{a} + \vec{b}}{2}$

(i) Midpoint of a line segment

(ii) Centroid of a triangle

(iii) Centroid of a tetrahedron

(iv) Point dividing AB internally 1:0

(v) Point dividing AB internally 0:1

(a) $\vec{b}$

(b) $\frac{\vec{a} + \vec{b}}{2}$

(c) $\frac{\vec{a} + \vec{b} + \vec{c}}{3}$

(d) $\frac{\vec{a} + \vec{b} + \vec{c} + \vec{d}}{4}$

(e) $\vec{a}$

(i) $\vec{r} = \frac{\vec{a} + \vec{b}}{2}$

(ii) $\vec{r} = \frac{2\vec{b} + \vec{a}}{3}$

(iii) $\vec{r} = 2\vec{b} - \vec{a}$

(iv) $\vec{r} = \vec{a} + 2(\vec{b}-\vec{a})$

(v) $\vec{r} = \vec{a} - (\vec{b}-\vec{a})$

(a) R divides AB internally in ratio 1:2.

(b) R divides AB externally in ratio 2:1.

(c) R is the midpoint of AB.

(d) R divides AB externally in ratio 1:2 (i.e., extends beyond A).

(e) R divides AB internally in ratio 2:1.

(i) Midpoint of BC

(ii) Centroid of triangle ABC

(iii) Point dividing median from A in ratio 2:1

(iv) Point dividing AB internally in ratio 1:1

(v) Point dividing AC internally in ratio 1:1

(a) $\frac{\vec{a} + \vec{c}}{2}$

(b) $\frac{\vec{a} + \vec{b} + \vec{c}}{3}$

(c) $\frac{\vec{b} + \vec{c}}{2}$

(d) Same as (ii)

(e) $\frac{\vec{a} + \vec{b}}{2}$

(i) R divides PQ in ratio $m:n$ internally

(ii) R divides QP in ratio $n:m$ internally

(iii) R divides PQ in ratio $m:n$ externally

(iv) R divides QP in ratio $n:m$ externally

(v) R is the midpoint of PQ

(a) $1:1$

(b) $\vec{r} = \frac{n\vec{p} + m\vec{q}}{m+n}$

(c) $\vec{r} = \frac{n\vec{q} + m\vec{p}}{n+m}$

(d) $\vec{r} = \frac{m\vec{q} - n\vec{p}}{m-n}$

(e) $\vec{r} = \frac{n\vec{p} - m\vec{q}}{n-m}$

(i) x-axis

(ii) xy-plane

(iii) Point (a,b,c)

(iv) Direction Cosines $(\ell,m,n)$

(v) Direction Ratios $(a,b,c)$

(a) $\ell^2 + m^2 + n^2 = 1$

(b) Equation $z=0$

(c) Proportional to direction cosines

(d) Equation $y=0, z=0$

(e) Has position vector $a\hat{i} + b\hat{j} + c\hat{k}$

(i) Vector form (point and parallel vector)

(ii) Vector form (two points)

(iii) Cartesian form (point and direction ratios)

(iv) Cartesian form (two points)

(v) Parametric Cartesian form

(a) $\frac{x-x_1}{x_2-x_1} = \frac{y-y_1}{y_2-y_1} = \frac{z-z_1}{z_2-z_1}$

(b) $x=x_1+\lambda a, y=y_1+\lambda b, z=z_1+\lambda c$

(c) $\vec{r} = (1-\lambda)\vec{a} + \lambda\vec{b}$

(d) $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$

(e) $\vec{r} = \vec{a} + \lambda \vec{b}$

(i) Distance between $(x_1, y_1, z_1)$ and origin

(ii) Distance between $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$

(iii) Distance between origin and point $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$

(iv) Magnitude of vector $\vec{PQ}$

(v) Distance between $(a,b,c)$ and the xy-plane

(a) $|c|$

(b) $\sqrt{x_1^2 + y_1^2 + z_1^2}$

(c) $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$

(d) Same as (ii)

(e) $\sqrt{x^1+y^1+z^1}$

(i) $\vec{a}$ in $\vec{r} = \vec{a} + \lambda \vec{b}$

(ii) $\vec{b}$ in $\vec{r} = \vec{a} + \lambda \vec{b}$

(iii) $\lambda$ in $\vec{r} = \vec{a} + \lambda \vec{b}$

(iv) $(a,b,c)$ in $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$

(v) $(x_1,y_1,z_1)$ in $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$

(a) A point on the line.

(b) Direction vector.

(c) Scalar parameter.

(d) Direction ratios.

(i) Direction cosines of a line

(ii) Direction ratios of a line

(iii) Sum of squares of direction cosines

(iv) Angles $\alpha, \beta, \gamma$ with axes

(v) Line makes equal angles with axes

(a) Can be any three numbers proportional to direction cosines.

(b) Value is 1.

(c) Cosines of these angles are direction cosines.

(d) Direction cosines are $(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}})$ or $(-\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}})$.

(e) Cosines of angles made with coordinate axes.

(i) Vector form (Normal form)

(ii) Vector form (Point-Normal form)

(iii) Cartesian form (General form)

(iv) Cartesian form (Intercept form)

(v) Cartesian form (Normal form)

(a) $Ax + By + Cz + D = 0$

(b) $lx + my + nz = p$ (where $\ell,m,n$ are direction cosines of normal, $p$ is distance from origin)

(c) $\vec{r} \cdot \hat{n} = d$

(d) $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$

(e) $(\vec{r} - \vec{a}) \cdot \vec{n} = 0$

(i) $2x - y + 3z = 5$

(ii) $\vec{r} \cdot (4\hat{i} + \hat{j} - 2\hat{k}) = 10$

(iii) Plane passing through origin

(iv) Plane parallel to xy-plane

(v) Normal vector to plane $Ax+By+Cz+D=0$

(a) $Ax+By+Cz=0$

(b) Normal vector is $2\hat{i} - \hat{j} + 3\hat{k}$

(c) Vector $A\hat{i} + B\hat{j} + C\hat{k}$

(d) Normal vector is $4\hat{i} + \hat{j} - 2\hat{k}$

(e) Equation form $z=c$ or $Az+D=0$ (with A=0, B=0)

(i) Plane through 3 points

(ii) Plane through a point perpendicular to a vector

(iii) Plane through origin and perpendicular to a vector

(iv) Plane through a point and containing two non-collinear vectors

(v) Plane defined by intercepts

(a) A point and a normal vector.

(b) Intercepts a, b, c on axes (non-zero).

(c) Three non-collinear points.

(d) Origin and a normal vector.

(e) A point and two non-collinear vectors parallel to the plane.

(i) $x+y+z=3$

(ii) $2x-y+z=0$

(iii) $x=5$

(iv) $z=0$

(v) $\vec{r} \cdot (\hat{i} + \hat{j}) = 2$

(a) (5, 1, 2)

(b) (1, 1, 1)

(c) (0, 0, 0)

(d) (1, 1, 0)

(e) (1, 2, 0)

(i) Components of normal vector $\vec{n}$

(ii) Unit normal vector $\hat{n}$

(iii) Distance from origin $d$ (when $D \neq 0$)

(iv) Sign of $d$ (distance from origin)

(v) Constant term $D$

(a) $\frac{|-D|}{\sqrt{A^2+B^2+C^2}}$

(b) $(A, B, C)$

(c) $\frac{A\hat{i} + B\hat{j} + C\hat{k}}{\sqrt{A^2+B^2+C^2}}$ if $D<0$ or $\frac{-A\hat{i} - B\hat{j} - C\hat{k}}{\sqrt{A^2+B^2+C^2}}$ if $D>0$.

(d) Related to distance and direction of normal.

(e) Must be positive in the normal form $lx+my+nz=p$ where $p>0$.

(i) Two lines

(ii) Two planes

(iii) A line and a plane

(iv) A line and a coordinate axis

(v) A plane and a coordinate plane

(a) Normal vector of plane and normal vector of coordinate plane.

(b) Direction vector of line and direction vector of coordinate axis.

(c) Direction vectors of the lines.

(d) Normal vectors of the planes.

(e) Direction vector of line and normal vector of plane (complementary angle).

(i) Two lines $\vec{b}_1 \cdot \vec{b}_2 = 0$

(ii) Two lines $\vec{b}_1 = k\vec{b}_2$

(iii) Two planes $\vec{n}_1 \cdot \vec{n}_2 = 0$

(iv) Two planes $\vec{n}_1 = k\vec{n}_2$

(v) Line $\vec{b}$ and Plane $\vec{n}$ with $\vec{b} \cdot \vec{n} = 0$

(a) Planes are parallel (or coincident).

(b) Lines are perpendicular.

(c) Line is parallel to the plane (or lies in the plane).

(d) Planes are perpendicular.

(e) Lines are parallel (or collinear).

(i) Cosine of angle between two lines (direction vectors $\vec{b}_1, \vec{b}_2$)

(ii) Cosine of angle between two planes (normal vectors $\vec{n}_1, \vec{n}_2$)

(iii) Sine of angle between a line $\vec{b}$ and a plane $\vec{n}$

(iv) Cosine of angle between a line $\vec{b}$ and the normal $\vec{n}$

(v) Sine of angle between a line $\vec{b}$ and the normal $\vec{n}$

(a) $\frac{|\vec{b} \cdot \vec{n}|}{|\vec{b}||\vec{n}|}$

(b) $\frac{|\vec{b} \cdot \vec{n}|}{|\vec{b}||\vec{n}|}$

(c) $\frac{|\vec{b} \cdot \vec{n}|}{|\vec{b}||\vec{n}|}$

(d) $\frac{|\vec{n}_1 \cdot \vec{n}_2|}{|\vec{n}_1||\vec{n}_2|}$

(e) $\frac{|\vec{b}_1 \cdot \vec{b}_2|}{|\vec{b}_1||\vec{b}_2|}$

(i) Lines $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ are parallel

(ii) Lines $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ are perpendicular

(iii) Planes $(A_1, B_1, C_1)$ and $(A_2, B_2, C_2)$ are parallel

(iv) Planes $(A_1, B_1, C_1)$ and $(A_2, B_2, C_2)$ are perpendicular

(v) Line $(a, b, c)$ and Plane $(A, B, C)$ are perpendicular

(a) $A_1A_2 + B_1B_2 + C_1C_2 = 0$

(b) $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

(c) $aA + bB + cC = 0$ (Line parallel to plane)

(d) $a_1a_2 + b_1b_2 + c_1c_2 = 0$

(e) $\frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2}$

(i) Line $\frac{x-x_1}{a_1}=\frac{y-y_1}{b_1}=\frac{z-z_1}{c_1}$ and Line $\frac{x-x_2}{a_2}=\frac{y-y_2}{b_2}=\frac{z-z_2}{c_2}$

(ii) Plane $A_1x+B_1y+C_1z+D_1=0$ and Plane $A_2x+B_2y+C_2z+D_2=0$

(iii) Line $\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}$ and Plane $Ax+By+Cz+D=0$ (use sine for line-plane angle)

(iv) x-axis and y-axis

(v) xy-plane and xz-plane

(a) Formula involves $\frac{|aA+bB+cC|}{\sqrt{a^2+b^2+c^2}\sqrt{A^2+B^2+C^2}}$ (This is cos of angle between line and normal)

(b) $\frac{|A_1A_2+B_1B_2+C_1C_2|}{\sqrt{A_1^2+B_1^2+C_1^2}\sqrt{A_2^2+B_2^2+C_2^2}}$

(c) 0

(d) $\frac{|a_1a_2+b_1b_2+c_1c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$

(e) 0

(i) Point to Plane

(ii) Point to Origin

(iii) Two Points

(iv) Two Parallel Planes

(v) Two Skew Lines

(a) Distance formula $\sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2}$

(b) $\frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2+B^2+C^2}}$

(c) $\frac{|D_2 - D_1|}{\sqrt{A^2+B^2+C^2}}$

(d) $\frac{|(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)|}{|\vec{b}_1 \times \vec{b}_2|}$

(e) $\sqrt{x^1+y^1+z^1}$

(i) Shortest distance between two skew lines $\vec{r} = \vec{a}_1 + \lambda \vec{b}_1, \vec{r} = \vec{a}_2 + \mu \vec{b}_2$

(ii) Shortest distance between two parallel lines $\vec{r} = \vec{a}_1 + \lambda \vec{b}, \vec{r} = \vec{a}_2 + \mu \vec{b}$

(iii) Shortest distance between two intersecting lines

(iv) Shortest distance between a point P and a line L (point A on L, direction $\vec{b}$)

(v) Shortest distance between a point P and itself

(a) 0

(b) $\frac{|(\vec{a}_2 - \vec{a}_1) \times \vec{b}|}{|\vec{b}|}$

(c) 0

(d) $\frac{|(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)|}{|\vec{b}_1 \times \vec{b}_2|}$

(e) $\frac{|\vec{AP} \times \vec{b}|}{|\vec{b}|}$

(i) Distance of (1,1,1) from $x+y+z=3$

(ii) Distance of (1,2,3) from $z=0$

(iii) Distance of origin from $x=5$

(iv) Distance of origin from $2x+3y+6z=7$

(v) Distance of (1,1,1) from $x=0$

(a) 3

(b) 1

(c) 0

(d) 5

(e) $\frac{7}{\sqrt{2^2+3^2+6^2}} = \frac{7}{\sqrt{4+9+36}} = \frac{7}{\sqrt{49}} = 1$

(i) Shortest distance is 0

(ii) Shortest distance is non-zero

(iii) Lines are coplanar

(iv) Lines are intersecting

(v) Lines are parallel and distinct

(a) Shortest distance formula for parallel lines is used.

(b) The lines are skew.

(c) The lines are intersecting or coincident.

(d) Shortest distance is 0.

(e) Shortest distance formula for skew lines is not applicable (denominator $\vec{b}_1 \times \vec{b}_2 = \vec{0}$).

(i) Distance between two parallel planes

(ii) Distance between two intersecting planes

(iii) Distance between coincident planes

(iv) Distance between plane $Ax+By+Cz+D=0$ and origin

(v) Distance between planes $x+y+z=1$ and $2x+2y+2z=2$

(a) 0

(b) $\frac{|D|}{\sqrt{A^2+B^2+C^2}}$

(c) Formula $\frac{|D_2-D_1|}{\sqrt{A^2+B^2+C^2}}$ applies.

(d) 0

(e) 0 (Planes are the same after simplification)