Vardaan Learning Institute

If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{b} = 2\hat{i} - \hat{j} + 3\hat{k}$, find a unit vector in the direction of $2\vec{a} - \vec{b}$.

Find a vector of magnitude 11 units in the direction opposite to $\vec{PQ}$, where $P(1,3,2)$ and $Q(-1,0,8)$ are two points.

Show that the points $A(1, -2, -8)$, $B(5, 0, -2)$, and $C(11, 3, 7)$ are collinear using vector algebra.

Let $\vec{a} = \hat{i} + 2\hat{j}$ and $\vec{b} = 2\hat{i} + \hat{j}$. Is $|\vec{a}| = |\vec{b}|$? Are the vectors $\vec{a}$ and $\vec{b}$ equal?

If $ABCD$ is a parallelogram and the position vectors of $A, B, C$ are $\hat{i}+\hat{j}$, $2\hat{i}+4\hat{j}$, and $3\hat{i}+5\hat{j}$ respectively, find the position vector of $D$.

Find the scalar and vector components of the vector with initial point $(2, 1)$ and terminal point $(-5, 7)$.

If $\vec{a} = \lambda\hat{i} + \hat{j} + 4\hat{k}$ and $\vec{b} = 2\hat{i} + 6\hat{j} + 24\hat{k}$ are collinear vectors, find the value of $\lambda$.

If $\vec{a}$ and $\vec{b}$ are the position vectors of $A$ and $B$, write the position vector of a point $C$ on $AB$ produced such that $\vec{AC} = 3\vec{AB}$.

Show that the vectors $2\hat{i} - 3\hat{j} + 4\hat{k}$ and $-4\hat{i} + 6\hat{j} - 8\hat{k}$ are collinear.

Find the unit vector parallel to the resultant of the vectors $\vec{a} = 2\hat{i} + 2\hat{j} - 5\hat{k}$ and $\vec{b} = 2\hat{i} + \hat{j} + 3\hat{k}$.

Find the direction cosines of a vector that makes equal acute angles with the positive directions of the coordinate axes.

If a line makes angles $90^\circ, 60^\circ$ and $30^\circ$ with the positive direction of $x, y$ and $z$ axes respectively, find its direction cosines.

Find the direction cosines of the vector $\vec{r} = 2\hat{i} - 3\hat{j} + 6\hat{k}$.

A vector $\vec{r}$ is inclined at equal angles to the three axes. If the magnitude of $\vec{r}$ is $2\sqrt{3}$ units, find $\vec{r}$.

Consider two points $P$ and $Q$ with position vectors $\vec{OP} = 3\vec{a} - 2\vec{b}$ and $\vec{OQ} = \vec{a} + \vec{b}$. Find the position vector of a point $R$ which divides the line joining $P$ and $Q$ internally in the ratio 2:1.

Find the position vector of point $R$ which divides the line joining $P$ and $Q$ (from Q15) externally in the ratio 2:1.

Show that the points $A(2\hat{i} - \hat{j} + \hat{k})$, $B(\hat{i} - 3\hat{j} - 5\hat{k})$, and $C(3\hat{i} - 4\hat{j} - 4\hat{k})$ form the vertices of a right-angled triangle.

Write the direction ratios of the vector $3\vec{a} + 2\vec{b}$ where $\vec{a} = \hat{i} + \hat{j} - 2\hat{k}$ and $\vec{b} = 2\hat{i} - 4\hat{j} + 5\hat{k}$.

Are the points with position vectors $60\hat{i} + 3\hat{j}$, $40\hat{i} - 8\hat{j}$, and $a\hat{i} - 52\hat{j}$ collinear? If yes, find $a$.

Find the position vector of the point $R$ which divides the line segment joining $A(\hat{i} + 2\hat{j} - \hat{k})$ and $B(-\hat{i} + \hat{j} + \hat{k})$ such that $2\vec{AR} = 3\vec{RB}$.

Find the angle between the vectors $\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}$ and $\vec{b} = 3\hat{i} - 2\hat{j} + \hat{k}$.

Find the projection of the vector $\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$ on the vector $\vec{b} = \hat{i} + 2\hat{j} + 2\hat{k}$.

Find the value of $\lambda$ if the vectors $\vec{a} = 2\hat{i} + \lambda\hat{j} + \hat{k}$ and $\vec{b} = \hat{i} - 2\hat{j} + 3\hat{k}$ are orthogonal.

Evaluate $|\vec{a}+\vec{b}|$ if it is given that $|\vec{a}|=3$, $|\vec{b}|=4$ and the angle between $\vec{a}$ and $\vec{b}$ is $60^\circ$.

If $\vec{a} + \vec{b} + \vec{c} = \vec{0}$, $|\vec{a}| = 3$, $|\vec{b}| = 5$, and $|\vec{c}| = 7$, find the angle between $\vec{a}$ and $\vec{b}$.

If $\vec{a}, \vec{b}, \vec{c}$ are mutually perpendicular vectors of equal magnitude, show that the vector $\vec{a} + \vec{b} + \vec{c}$ is equally inclined to $\vec{a}, \vec{b}$, and $\vec{c}$.

Find the value of $p$ for which the vectors $3\hat{i} + 2\hat{j} + 9\hat{k}$ and $\hat{i} + p\hat{j} + 3\hat{k}$ are perpendicular.

Show that $(\vec{a} - \vec{b}) \cdot (\vec{a} + \vec{b}) = 0$ implies $|\vec{a}| = |\vec{b}|$.

If $\vec{a}$ is a unit vector and $(\vec{x} - \vec{a}) \cdot (\vec{x} + \vec{a}) = 15$, find $|\vec{x}|$.

If $|\vec{a}| = a$ and $|\vec{b}| = b$, prove that $\left(\frac{\vec{a}}{a^2} - \frac{\vec{b}}{b^2}\right)^2 = \left(\frac{\vec{a} - \vec{b}}{ab}\right)^2$.

Find the vector projection of $\vec{a} = \hat{i} - \hat{j}$ on $\vec{b} = \hat{i} + \hat{j}$.

Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors such that $|\vec{a}|=3, |\vec{b}|=4, |\vec{c}|=5$ and each one of them is perpendicular to the sum of the other two. Find $|\vec{a}+\vec{b}+\vec{c}|$.

Find a unit vector perpendicular to both $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} + 3\hat{k}$.

Find the area of the parallelogram whose adjacent sides are determined by the vectors $\vec{a} = \hat{i} - \hat{j} + 3\hat{k}$ and $\vec{b} = 2\hat{i} - 7\hat{j} + \hat{k}$.

Find the area of a triangle with vertices $A(1, 1, 2)$, $B(2, 3, 5)$, and $C(1, 5, 5)$.

If $\vec{a} \times \vec{b} = \vec{c} \times \vec{d}$ and $\vec{a} \times \vec{c} = \vec{b} \times \vec{d}$, show that $\vec{a} - \vec{d}$ is parallel to $\vec{b} - \vec{c}$ (given $\vec{a} \neq \vec{d}$ and $\vec{b} \neq \vec{c}$).

Evaluate $|\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2$ if $|\vec{a}| = 2$ and $|\vec{b}| = 5$.

Find the area of the parallelogram whose diagonals are represented by the vectors $\vec{d_1} = 2\hat{i} - \hat{j} + \hat{k}$ and $\vec{d_2} = 3\hat{i} + 4\hat{j} - \hat{k}$.

Find a vector of magnitude 5 units, perpendicular to each of the vectors $(\vec{a}+\vec{b})$ and $(\vec{a}-\vec{b})$, where $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} + 3\hat{k}$.

Prove that $(\vec{a} \times \vec{b})^2 = |\vec{a}|^2 |\vec{b}|^2 - (\vec{a} \cdot \vec{b})^2$. (Lagrange's Identity)

If $\hat{i}, \hat{j}, \hat{k}$ are mutually perpendicular unit vectors, evaluate $(2\hat{i} - 3\hat{j}) \times (3\hat{i} + 4\hat{j})$.

Given $|\vec{a}| = 10, |\vec{b}| = 2$ and $\vec{a} \cdot \vec{b} = 12$, find $|\vec{a} \times \vec{b}|$.

Vectors $\vec{a}, \vec{b}, \vec{c}$ are such that $\vec{a} + \vec{b} + \vec{c} = \vec{0}$. Prove that $\vec{a} \times \vec{b} = \vec{b} \times \vec{c} = \vec{c} \times \vec{a}$.

A force $\vec{F} = 2\hat{i} + \hat{j} - \hat{k}$ acts at a point $P(1, 2, 3)$. Find the torque (moment of force) about the origin.

Find the volume of the parallelepiped whose coterminous edges are $\vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k}$, $\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$, and $\vec{c} = 3\hat{i} - \hat{j} + 2\hat{k}$.

Show that the vectors $\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}$, $\vec{b} = -2\hat{i} + 3\hat{j} - 4\hat{k}$, and $\vec{c} = \hat{i} - 3\hat{j} + 5\hat{k}$ are coplanar.

Find $\lambda$ for which the vectors $2\hat{i} - \hat{j} + \hat{k}$, $\hat{i} + 2\hat{j} - 3\hat{k}$, and $3\hat{i} + \lambda\hat{j} + 5\hat{k}$ are coplanar.

Prove that $[\vec{a}+\vec{b}, \vec{b}+\vec{c}, \vec{c}+\vec{a}] = 2[\vec{a} \, \vec{b} \, \vec{c}]$.

Prove that the four points $A(4,5,1), B(0,-1,-1), C(3,9,4)$, and $D(-4,4,4)$ are coplanar.

Simplify the expression: $\vec{i} \cdot (\vec{j} \times \vec{k}) + \vec{j} \cdot (\vec{i} \times \vec{k}) + \vec{k} \cdot (\vec{i} \times \vec{j})$.

Using the BAC-CAB rule, expand $\vec{a} \times (\vec{b} \times \vec{c}) + \vec{b} \times (\vec{c} \times \vec{a}) + \vec{c} \times (\vec{a} \times \vec{b})$.

Evaluate $\vec{a} \times (\vec{a} \times \vec{b})$ if $\vec{a} = \hat{i} + \hat{j}$ and $\vec{b} = \hat{i} - \hat{j}$.

If $[\vec{a} \, \vec{b} \, \vec{c}] = 3$, find the value of $[\vec{a} - \vec{b}, \vec{b} - \vec{c}, \vec{c} - \vec{a}]$.

Show that $[\vec{a}\times\vec{b}, \vec{b}\times\vec{c}, \vec{c}\times\vec{a}] = [\vec{a} \, \vec{b} \, \vec{c}]^2$.

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors, prove that $\frac{\vec{a} \cdot (\vec{b} \times \vec{c})}{(\vec{c} \times \vec{a}) \cdot \vec{b}} + \frac{\vec{b} \cdot (\vec{a} \times \vec{c})}{\vec{c} \cdot (\vec{a} \times \vec{b})} = 0$.