Vardaan Learning Institute

Which of the following physical quantities is a vector?
(a) Electric Current (b) Pressure (c) Angular Velocity (d) Electric Potential

Find the magnitude of the vector $\vec{A} = 3\hat{i} + 4\hat{j} + 12\hat{k}$.

Determine the unit vector parallel to the resultant of the vectors $\vec{P} = 2\hat{i} + 4\hat{j} - 5\hat{k}$ and $\vec{Q} = \hat{i} + 2\hat{j} + 3\hat{k}$.

If $0.4\hat{i} + 0.8\hat{j} + c\hat{k}$ represents a unit vector, find the value of $c$.

Two vectors are given as $\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}$ and $\vec{b} = x\hat{i} + y\hat{j} + z\hat{k}$. If $\vec{a}$ and $\vec{b}$ are equal vectors, write the values of $x, y,$ and $z$.

A point $P$ has coordinates $(2, -1, 4)$. Write its position vector with respect to the origin.

Calculate the displacement vector from point $A(1, 2, 3)$ to point $B(4, 5, 6)$ and find its magnitude.

What is a null (or zero) vector? State two physical situations where a null vector arises.

Are the vectors $\vec{A} = 2\hat{i} + 3\hat{j} - 4\hat{k}$ and $\vec{B} = 4\hat{i} + 6\hat{j} - 8\hat{k}$ collinear? Justify mathematically.

State the difference between polar vectors and axial vectors, giving one example of each.

The maximum and minimum magnitudes of the resultant of two vectors are $17$ units and $7$ units respectively. Find the magnitude of each vector.

Two forces, each of magnitude $F$, act on a particle. If their resultant is also equal to $F$, find the angle between the two forces.

If $|\vec{A} + \vec{B}| = |\vec{A} - \vec{B}|$, what is the angle between vectors $\vec{A}$ and $\vec{B}$?

Two forces of $5 \text{ N}$ and $12 \text{ N}$ act on a particle at an angle of $90^\circ$. Calculate the magnitude and direction of the resultant force.

Prove that $|\vec{A} + \vec{B}|^2 + |\vec{A} - \vec{B}|^2 = 2(A^2 + B^2)$.

Three vectors $\vec{A}, \vec{B}, \vec{C}$ satisfy the relation $\vec{A} + \vec{B} + \vec{C} = 0$. If $|\vec{A}| = 3$, $|\vec{B}| = 4$, and $|\vec{C}| = 5$, find the angle between $\vec{A}$ and $\vec{B}$.

A car travels $10 \text{ km}$ North, then $10\sqrt{2} \text{ km}$ South-West. What is the net displacement of the car from the starting point?

Two vectors have magnitudes $A$ and $B$. Under what conditions will their resultant have a magnitude of $(A^2 + B^2 + AB)^{1/2}$?

A particle is moving eastwards with a velocity of $5 \text{ m/s}$. In $10 \text{ s}$, the velocity changes to $5 \text{ m/s}$ northwards. Find the average acceleration.

Given $\vec{a} + \vec{b} + \vec{c} + \vec{d} = 0$. Which of the following statements must be true? (a) $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ must be coplanar. (b) The magnitude of $\vec{a}$ can never be greater than the sum of magnitudes of $\vec{b}, \vec{c}$ and $\vec{d}$.

Find the resultant of two forces $P$ and $2P$ acting at an angle of $120^\circ$.

If $\hat{n}_1$ and $\hat{n}_2$ are two unit vectors and $\theta$ is the angle between them, prove that $\sin(\theta/2) = \frac{1}{2} |\hat{n}_1 - \hat{n}_2|$.

A force of $100 \text{ N}$ makes an angle of $60^\circ$ with the X-axis. Find its X and Y components.

Write a vector of magnitude 10 units which makes an angle of $37^\circ$ with the positive X-axis in the XY plane.

Find the direction cosines $l, m, n$ of the vector $\vec{R} = 2\hat{i} - 3\hat{j} + 6\hat{k}$.

If a vector makes angles $\alpha, \beta, \gamma$ with the coordinate axes, prove that $\sin^2\alpha + \sin^2\beta + \sin^2\gamma = 2$.

A block of mass $M$ is placed on a smooth inclined plane of angle $\theta$. Resolve its weight into components parallel and perpendicular to the incline.

What is the angle made by the vector $\vec{A} = \hat{i} + \hat{j} + \sqrt{2}\hat{k}$ with the Z-axis?

Find the vector projection (component) of $\vec{A} = 2\hat{i} + 3\hat{j}$ along the vector $\vec{B} = \hat{i} + \hat{j}$.

A force is inclined at $60^\circ$ to the horizontal. If its rectangular horizontal component is $50 \text{ N}$, find the magnitude of the force and its vertical component.

If $\vec{A} = 3\hat{i} + 4\hat{j}$ and $\vec{B} = 7\hat{i} + 24\hat{j}$, find a vector having the same magnitude as $\vec{B}$ and parallel to $\vec{A}$.

Can a 3D vector have direction angles $45^\circ, 60^\circ,$ and $120^\circ$ with the x, y, and z axes respectively? Justify your answer.

Calculate the scalar product of $\vec{A} = 2\hat{i} - 5\hat{j} + 3\hat{k}$ and $\vec{B} = \hat{i} + 2\hat{j} - \hat{k}$.

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

Determine the value of '$m$' for which the vectors $\vec{A} = 3\hat{i} - 2\hat{j} + m\hat{k}$ and $\vec{B} = 2\hat{i} + \hat{j} + 4\hat{k}$ are mutually perpendicular.

A constant force $\vec{F} = (2\hat{i} + 3\hat{j} + 4\hat{k}) \text{ N}$ displaces a particle from $(1, 1, 1) \text{ m}$ to $(2, 3, 5) \text{ m}$. Calculate the work done by the force.

Evaluate: $(\vec{A} + \vec{B}) \cdot (\vec{A} - \vec{B})$. What is the physical meaning if this dot product equals zero?

If $\vec{A}$ and $\vec{B}$ are two unit vectors such that $\vec{A} + \vec{B}$ is also a unit vector, find the angle between them using dot product properties.

The power of a motor is given by $P = \vec{F} \cdot \vec{v}$. Find the power if $\vec{F} = 10\hat{i} + 10\hat{j} + 20\hat{k}$ and velocity $\vec{v} = 5\hat{i} - 3\hat{j} + 6\hat{k}$.

Find the scalar projection of vector $\vec{a} = 3\hat{i} - \hat{j} - 2\hat{k}$ on the vector $\vec{b} = \hat{i} + 2\hat{j} - 3\hat{k}$.

Show that the vectors $\vec{a} = \frac{1}{7}(2\hat{i} + 3\hat{j} + 6\hat{k})$, $\vec{b} = \frac{1}{7}(3\hat{i} - 6\hat{j} + 2\hat{k})$, and $\vec{c} = \frac{1}{7}(6\hat{i} + 2\hat{j} - 3\hat{k})$ are mutually perpendicular unit vectors.

If $\vec{A} \cdot \vec{B} = |\vec{A} \times \vec{B}|$, what is the angle between $\vec{A}$ and $\vec{B}$?

Determine if the triangle formed by position vectors $\vec{r}_1 = \hat{i} - 2\hat{j} + 3\hat{k}$, $\vec{r}_2 = 2\hat{i} + 3\hat{j} - 4\hat{k}$, and $\vec{r}_3 = -7\hat{j} + 10\hat{k}$ is a right-angled triangle.

Find the flux ($\Phi = \vec{E} \cdot \vec{A}$) of a uniform electric field $\vec{E} = 2\times 10^3 \hat{i} \text{ N/C}$ through a square area $\vec{A}$ of $100 \text{ cm}^2$ lying in the Y-Z plane.

Calculate the vector product $\vec{A} \times \vec{B}$ if $\vec{A} = 3\hat{i} + 2\hat{j} - \hat{k}$ and $\vec{B} = \hat{i} - \hat{j} + \hat{k}$.

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

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

Prove that if $\vec{A} \times \vec{B} = 0$ (and neither is a null vector), then the vectors are parallel. What is the value of $\vec{A} \cdot \vec{B}$ in this case?

Calculate the torque ($\vec{\tau} = \vec{r} \times \vec{F}$) produced by a force $\vec{F} = 3\hat{i} + \hat{j} + 5\hat{k}$ acting at a point whose position vector is $\vec{r} = 2\hat{i} + 3\hat{j} - \hat{k}$.

Determine the area of a triangle formed by the vectors $\vec{A} = 3\hat{i} + 4\hat{j}$ and $\vec{B} = -3\hat{i} + 7\hat{j}$ as its adjacent sides.

Simplify the expression: $(\vec{A} - \vec{B}) \times (\vec{A} + \vec{B})$.

Find the value of $p$ such that vectors $\vec{a} = 2\hat{i} + 3\hat{j} - \hat{k}$ and $\vec{b} = -4\hat{i} - 6\hat{j} + p\hat{k}$ are parallel.

Evaluate: $\hat{i} \cdot (\hat{j} \times \hat{k}) + \hat{j} \cdot (\hat{i} \times \hat{k}) + \hat{k} \cdot (\hat{i} \times \hat{j})$.

Show that $|\vec{A} \times \vec{B}|^2 + (\vec{A} \cdot \vec{B})^2 = |\vec{A}|^2 |\vec{B}|^2$. (Lagrange's Identity)

A particle has angular velocity $\vec{\omega} = \hat{i} - 2\hat{j} + 2\hat{k}$ and position vector $\vec{r} = 0.4\hat{j} - 0.3\hat{k}$. Find its linear velocity $\vec{v}$ (given $\vec{v} = \vec{\omega} \times \vec{r}$).