Vector Algebra

The direction ratios of the vector $\hat{k}$ are:

If $\vec{a}$ is a vector, then $|-\vec{a}|$ is equal to:

The magnitude of the vector $\sqrt{2}\hat{i} + \sqrt{2}\hat{j}$ is:

The scalar components of the vector joining $A(2, 3, 0)$ and $B(-1, -2, 0)$ are:

A vector $\vec{v}$ has magnitude $12$ units and is inclined at an angle of $45^\circ$ with the positive $x$-axis and $120^\circ$ with the positive $z$-axis. If the scalar component of $\vec{v}$ along the $y$-axis is positive, then its value is:

Find the position vector of the point $R$ which divides the line joining $P(\hat{i} + 2\hat{j} - \hat{k})$ and $Q(-\hat{i} + \hat{j} + \hat{k})$ in the ratio $2:1$ internally.

If $\vec{a} = \hat{i} + \hat{j}$, $\vec{b} = \hat{j} + \hat{k}$ and $\vec{c} = \hat{k} + \hat{i}$, then find the magnitude of $\vec{a} + \vec{b} + \vec{c}$.

Let $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$. If $\vec{b}$ is a vector such that $\vec{a} \cdot \vec{b} = |\vec{a}|^2$ and $|\vec{b} - \vec{a}| = \sqrt{7}$, then $|\vec{b}|$ is:

If the angle between the vector $\vec{a}$ and the positive $x$-axis is $45^\circ$ and with the positive $y$-axis is $60^\circ$, then the angle with the $z$-axis cannot be:

If $\cos \alpha, \cos \beta, \cos \gamma$ are direction cosines of a vector, find the value of $\cos 2\alpha + \cos 2\beta + \cos 2\gamma + 1$.

What are the projections of the vector $\vec{a} = 2\hat{i} + 3\hat{j} + 6\hat{k}$ on the coordinate axes?

A point $P$ is at a distance of $1$ unit from the origin. If the line $OP$ makes an angle of $45^\circ$ with both the $x$ and $y$ axes and $90^\circ$ with the $z$ axis, the coordinates of $P$ are:

The direction ratios of the line $\frac{x}{2} = \frac{y}{3} = \frac{z}{4}$ are:

If a line makes an angle $\alpha$ with each of the $x, y$, and $z$ axes, then $\cos^2 \alpha$ is:

Direction ratios of the normal to the $yz$-plane are:

The angle between a line with direction ratios $(1, 1, 0)$ and the $z$-axis is:

Find the direction ratios of a vector $\vec{a}$ if $\vec{a}$ is equally inclined to the axes and its magnitude is $\sqrt{3}$.

If the coordinates of the points $P$ and $Q$ are $(2, 3, 5)$ and $(4, 6, -1)$, then the direction cosines of the line segment $PQ$ are:

If a line makes an angle of $\pi/3$ with the $x$-axis and $\pi/4$ with the $y$-axis, the angle it makes with the $z$-axis is:

A line segment $AB$ of length $r$ has direction cosines $(l, m, n)$. If the coordinates of $A$ are $(x_1, y_1, z_1)$, the coordinates of $B$ are:

The vector components of $\vec{r} = -2\hat{i} + 4\hat{k}$ are:

Which of the following describes the vector $\hat{i} - \hat{i}$?

If $\vec{a}$ is a vector of magnitude 5, then the magnitude of $-3\vec{a}$ is:

The vector $\vec{PQ}$ where $P$ is $(1, 2, 3)$ and $Q$ is $(1, 2, 10)$ is:

What is the scalar component of the vector $\vec{a} = 3\hat{i} - 4\hat{j} + 12\hat{k}$ along the $z$-axis?

The displacement vector of a particle moving from point $A(3, -2, 1)$ to $B(1, 2, 3)$ is:

Find the position vector of the point which divides the join of $P(2\vec{a} + \vec{b})$ and $Q(\vec{a} - 2\vec{b})$ in the ratio $1:2$ internally.

The direction cosines of the vector $2\hat{i} + 2\hat{j} - \hat{k}$ are:

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

The vector having initial point $(2, 5, 0)$ and terminal point $(-1, -2, 4)$ is:

If $\vec{a} = 2\hat{i} - \hat{j}$, then the magnitude of the vector $3\vec{a}$ is:

The magnitude of the vector $\hat{i} + \hat{j} + \hat{k}$ is:

Two vectors are said to be equal if they have:

If $\vec{a} = \hat{i} + \hat{j}$, $\vec{b} = \hat{j} + \hat{k}$, and $\vec{c} = \hat{k} + \hat{i}$, then $\vec{a} + \vec{b} + \vec{c}$ is:

If $\vec{v} = \vec{a} - \vec{b} + \vec{c}$ and $\vec{a} = 2\hat{i} + \hat{j}$, $\vec{b} = \hat{i} - \hat{j}$, $\vec{c} = \hat{k}$, then $|\vec{v}|$ is:

The displacement vector from $A(3, -2, 1)$ to $B(1, 2, -3)$ is multiplied by a scalar $k=1/2$. The resulting vector is:

If $\vec{a} = 2\hat{i} + 3\hat{j} - 6\hat{k}$, then a vector in the direction of $\vec{a}$ with magnitude $21$ is:

If $\vec{a}$ and $\vec{b}$ are two vectors such that $|\vec{a}| = 2, |\vec{b}| = 1$ and $|\vec{a}-\vec{b}| = \sqrt{3}$, then the angle between $\vec{a}$ and $\vec{b}$ is:

In $\triangle ABC$, $P$ is a point on $BC$ such that $\vec{BP} = 2\vec{PC}$. If $\vec{AB} = \vec{c}$ and $\vec{AC} = \vec{b}$, then $\vec{AP}$ is:

If $\vec{a}$ and $\vec{b}$ are non-collinear vectors such that $\vec{c} = (x-2)\vec{a} + \vec{b}$ and $\vec{d} = (2x+1)\vec{a} - \vec{b}$ are collinear, then $x$ is:

Calculate the dot product of $\hat{i} - 2\hat{j} + \hat{k}$ and $\hat{i} + \hat{j} + \hat{k}$.

Find the value of $3\hat{k} \cdot 2\hat{k}$.

Find $\vec{a} \cdot \vec{b}$ if $\vec{a} = \hat{i} + \hat{k}$ and $\vec{b} = \hat{i} - \hat{k}$.

Evaluate the scalar product: $(\hat{i} - \hat{j}) \cdot (\hat{j} - \hat{k})$.

Let $\vec{u}$ and $\vec{v}$ be two vectors such that $|\vec{u}| = 3$, $|\vec{v}| = 2$ and $\vec{u} \cdot \vec{v} = 6$. What is the angle between $\vec{u}$ and $\vec{v}$?

If $|\vec{a}| = 3$, $|\vec{b}| = 4$, and the angle between them is $120^\circ$, find the value of $|2\vec{a} + \vec{b}|$.

If the projection of the vector $\vec{a} = \lambda\hat{i} + \hat{j} + 4\hat{k}$ on the vector $\vec{b} = 2\hat{i} + 6\hat{j} + 3\hat{k}$ is $4$ units, find the value of $\lambda$.

If $\vec{a}$ and $\vec{b}$ are unit vectors and $|\vec{a} + \vec{b}| = 1$, find the angle between $\vec{a}$ and $\vec{b}$.

If $\vec{a} = \hat{i} + 4\hat{j} + 2\hat{k}$, $\vec{b} = 3\hat{i} - 2\hat{j} + 7\hat{k}$ and $\vec{c} = 2\hat{i} - \hat{j} + 4\hat{k}$, find a vector $\vec{d}$ which is perpendicular to both $\vec{a}$ and $\vec{b}$ and $\vec{c} \cdot \vec{d} = 15$.

Find the work done by a force $\vec{F} = \hat{i} + 2\hat{j} + 3\hat{k}$ in moving a particle from point $A(1, 2, 3)$ to $B(2, 3, 5)$.

If $\vec{a} \times \vec{b} = \vec{a} \times \vec{c}$ for a non-zero vector $\vec{a}$, then:

The vector product of two non-zero vectors is always a:

What is the magnitude of $\hat{i} \times \hat{k}$?

Simplify: $\vec{a} \times (\vec{b} + \vec{c}) + \vec{b} \times (\vec{c} + \vec{a}) + \vec{c} \times (\vec{a} + \vec{b})$.

The unit vector perpendicular to both $\hat{i} + \hat{j}$ and $\hat{i} - \hat{j}$ is:

If $\vec{a}$ and $\vec{b}$ represent the sides of a rhombus, what is the value of $\vec{a} \times \vec{b}$ if the angle between them is $60^\circ$ and $|\vec{a}| = 2$?

Find the vector product of $\vec{a} = \hat{i} - 7\hat{j} + 7\hat{k}$ and $\vec{b} = 3\hat{i} - 2\hat{j} + 2\hat{k}$.

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three vectors such that $|\vec{a}| = 1, |\vec{b}| = 2$ and $|\vec{c}| = 3$. If the projection of $\vec{b}$ on $\vec{a}$ is equal to the projection of $\vec{c}$ on $\vec{a}$, and the vectors $\vec{b}$ and $\vec{c}$ are perpendicular to each other, then the value of $|\vec{a} \times (\vec{b} - \vec{c})|$ is:

If $\vec{a}$ and $\vec{b}$ are unit vectors and $\theta$ is the angle between them, find $|(\vec{a} + \vec{b}) \times (\vec{a} - \vec{b})|$.

If $\vec{a}, \vec{b}$ are two unit vectors such that $|\vec{a} \times \vec{b}| = 1$, then the angle between $\vec{a}$ and $\vec{b}$ is:

In the vector $5\hat{i} - 2\hat{j} + 7\hat{k}$, the number $7$ is called the:

If $|\vec{a}| = 0$, what can be said about $\vec{a}$?

Which of the following describes the Triangle Law of vector addition for vectors $\vec{AB}, \vec{BC}, \vec{AC}$?

The position vector of a point $A$ is $\vec{a}$ and $B$ is $\vec{b}$. The vector $\vec{BA}$ is:

Find the value of $\lambda + \mu$ if the vectors $2\hat{i} + 6\hat{j} + 27\hat{k}$ and $\hat{i} + \lambda\hat{j} + \mu\hat{k}$ are collinear.

If the magnitude of a vector is 0, what is the vector called?

Calculate the magnitude of the vector $\vec{a} = \hat{i} + 2\hat{j} - 2\hat{k}$.

The points with position vectors $60\hat{i} + 3\hat{j}$, $40\hat{i} - 8\hat{j}$, $a\hat{i} - 52\hat{j}$ are collinear. Find the value of $a$.

If the sum of unit vectors $\hat{i}, \hat{j}, \hat{k}$ is $\vec{s}$, find the magnitude of $\vec{s}$.

Find the length of the median through vertex $A$ of the triangle with vertices $A(2, 3, 5)$, $B(1, 0, 3)$ and $C(7, 2, 1)$.

Find the vector joining $A(1, 1, 1)$ and $B(2, 1, 1)$.

Calculate the magnitude of the vector joining $(1, 2, 3)$ and $(4, 6, 15)$.

Find the vector $\vec{PQ}$ for $P(1, 0, -1)$ and $Q(-1, 0, 1)$.

What is the magnitude of the vector joining $(0, 0, 0)$ and $(1, 0, 0)$?

Find the unit vector in the direction of $\vec{PQ}$ if $P = (1, 2)$ and $Q = (4, 6)$.