Vectors

Which of the following physical quantities is NOT a vector?

According to the triangle law of addition, if $\vec{PQ}$ and $\vec{QR}$ are two vectors, their sum is:

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

If a vector makes angles of $60^{\circ}, 60^{\circ}$ and $\gamma$ with the $x, y$ and $z$ axes, what is the value of $\gamma$?

In $\triangle ABC$, the position vectors of the vertices are $\vec{a} = \hat{j} + \hat{k}$, $\vec{b} = 2\hat{i} + 4\hat{j} + 3\hat{k}$, and $\vec{c} = 4\hat{i} + 2\hat{j} + \hat{k}$. If $M$ is the midpoint of the side $BC$, the magnitude of the median vector $\vec{AM}$ is:

Find the scalar components of a vector with initial point $P(2, 3, 0)$ and terminal point $Q(-1, -2, 4)$.

If the direction cosines of a line are $k, k, k$, then the value of $k$ is:

If $\vec{a} = 2\hat{i} + \lambda\hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} + 3\hat{k}$ are orthogonal, then $\lambda$ is:

If $\vec{a}$ is a unit vector and $(\vec{x} - \vec{a}) \cdot (\vec{x} + \vec{a}) = 8$, then $|\vec{x}|$ is:

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

What is the direction of the vector $-3\hat{k}$?

If $\vec{a} = 3\hat{i} - 4\hat{j}$, what is its magnitude?

Which of the following is true for the direction cosines of a vector?

The additive identity for vector addition is:

The value of $(\vec{a} \times \vec{b})^2$ is equal to:

Evaluate the scalar triple product $[2\hat{i} \quad \hat{j} \quad \hat{k}]$.

Find the vector $\vec{c}$ such that $\vec{a} \times \vec{c} = \vec{b}$ and $\vec{a} \cdot \vec{c} = 3$, where $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{b} = \hat{j} - \hat{k}$.

If $\vec{a}, \vec{b}, \vec{c}$ are vectors such that $[\vec{a} \vec{b} \vec{c}] = 4$, then $[\vec{a} \times \vec{b} \quad \vec{b} \times \vec{c} \quad \vec{c} \times \vec{a}]$ is:

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors and $\vec{d} = \lambda \vec{a} + \mu \vec{b} + \nu \vec{c}$, then $\nu$ is equal to:

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three vectors having magnitudes 1, 1 and 2 respectively. If $\vec{a} \times (\vec{a} \times \vec{c}) + \vec{b} = \vec{0}$, then the acute angle between $\vec{a}$ and $\vec{c}$ is:

If a line makes an angle of $180^\circ$ with the $X$-axis, its direction cosine $l$ is:

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

If a line is perpendicular to the $Y$-axis, its direction cosine $m$ is:

What are the direction cosines of the vector $-3\hat{i} + 4\hat{j}$?

If the direction cosines $l, m, n$ of two lines satisfy the relations $l + m + n = 0$ and $l^2 + m^2 - n^2 = 0$, then the measure of the acute angle between these two lines is:

The direction cosines of the vector $3\hat{i} - 4\hat{j} + 5\hat{k}$ are:

If $l, m, n$ are the direction cosines of a line, then the value of $l^2 + m^2$ is:

A line with direction ratios $(2, -1, 3)$ is reflected in a plane whose normal has direction ratios $(1, 2, -2)$. Which of the following sets of direction ratios correctly represents the reflected line?

The direction ratios of the line passing through $(2, 5, -3)$ and perpendicular to the plane $x-2y+z=7$ are:

If the direction cosines of a line are $l, m, n$, then the maximum value of $l+m+n$ is:

What is the angle between the vector $\vec{a}$ and the vector $5\vec{a}$ (where $\vec{a} \neq \vec{0}$)?

If $\vec{a} = \hat{i} + \hat{j}$, find the value of $\vec{a} \cdot \vec{a}$.

The expression $\frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}$ represents:

Find $(3\hat{i} - 2\hat{j}) \cdot (2\hat{i} + 3\hat{j})$.

If $|\vec{a}| = 1, |\vec{b}| = 2, |\vec{c}| = 3$ and $\vec{a}, \vec{b}, \vec{c}$ are mutually perpendicular, what is the value of $(\vec{a} + \vec{b} + \vec{c}) \cdot \vec{a}$?

Find the angle between the vectors $\vec{a} = 2\hat{i} + 2\hat{j} - \hat{k}$ and $\vec{b} = 6\hat{i} - 3\hat{j} + 2\hat{k}$.

If the projection of vector $\vec{a}$ on $\vec{b}$ is equal to the projection of $\vec{b}$ on $\vec{a}$, then:

If $\vec{a}$ and $\vec{b}$ are unit vectors such that $\vec{a} + \vec{b}$ is also a unit vector, find the angle between $\vec{a}$ and $\vec{b}$.

Let $\vec{a} = \hat{i} + \hat{j}$, $\vec{b} = \hat{j} + \hat{k}$ and $\vec{c} = x\vec{a} + y\vec{b}$. If the scalar projection of $\vec{c}$ on the vector $\vec{a} \times \vec{b}$ is zero, then:

If the scalar product of the vector $\hat{i} + \hat{j} + \hat{k}$ with the unit vector along the sum of vectors $2\hat{i} + 4\hat{j} - 5\hat{k}$ and $\lambda\hat{i} + 2\hat{j} + 3\hat{k}$ is equal to 1, find $\lambda$.

The value of $(\vec{a} \times \vec{b}) \cdot (\vec{a} + \vec{b})$ is:

If $(\vec{a} \times \vec{b}) \cdot \vec{c} = 0$, then the vectors $\vec{a}, \vec{b}, \vec{c}$ are:

If $\vec{a} \times \vec{b}$ is a unit vector, then $|\vec{a}||\vec{b}|\sin\theta$ must be:

Calculate $\hat{k} \times (\hat{j} \times \hat{i})$.

If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{b} = \hat{j} - \hat{k}$, find a vector $\vec{c}$ such that $\vec{a} \times \vec{c} = \vec{b}$ and $\vec{a} \cdot \vec{c} = 3$.

The area of the triangle formed by the points with position vectors $\hat{i}, \hat{j}, \hat{k}$ is:

Find $|\vec{a} \times \vec{b}|$ if $|\vec{a}| = 2, |\vec{b}| = 5$ and $\vec{a} \cdot \vec{b} = 1$.

Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that the angle between them is $60^\circ$. If a vector $\vec{c}$ is defined as $\vec{c} = \vec{a} - \vec{b} + 2(\vec{a} \times \vec{b})$, then the magnitude of $\vec{c}$ is:

Let $\vec{a} = \hat{i} + 2\hat{j} - \hat{k}$ and $\vec{b} = \hat{i} + \hat{j} + \hat{k}$. If $\vec{c}$ is a vector such that $\vec{c} \cdot \vec{a} = 8$ and $\vec{c} \times \vec{a} = \vec{b} \times \vec{a}$, then the value of the scalar product $\vec{c} \cdot \vec{b}$ is equal to:

If the volume of a parallelepiped formed by vectors $\vec{a}, \vec{b}, \vec{c}$ is 12, then the volume of the parallelepiped formed by vectors $\vec{a} \times \vec{b}, \vec{b} \times \vec{c}, \vec{c} \times \vec{a}$ is:

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

The scalar projection of $\vec{a} = 3\hat{i} + 4\hat{j} + 12\hat{k}$ on itself is:

The scalar projection of $\vec{a} = 2\hat{i} + 2\hat{j} + \hat{k}$ on itself is:

What is the projection of the vector $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$ on itself?

In $\triangle ABC$, the coordinates of the vertices are $A(1, 2, 1)$, $B(2, 0, 0)$, and $C(0, 2, 2)$. Let $M$ be the midpoint of side $BC$. The vector projection of the median vector $\vec{AM}$ on the side vector $\vec{BC}$ is given by:

A line passes through the origin and makes angles of $45^\circ$ and $60^\circ$ with the positive $x$ and $y$ axes, respectively. If its direction cosine with the $z$-axis is positive, find the scalar projection of the vector $\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$ on this line.

Find the scalar projection of $\vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k}$ on $\vec{b} = \hat{i} + 2\hat{j} + 2\hat{k}$.

If $\vec{a} = \hat{i} + k\hat{j} + \hat{k}$ and $\vec{b} = 4\hat{i} - 4\hat{j} + 7\hat{k}$ and the scalar projection of $\vec{a}$ on $\vec{b}$ is $\frac{19}{9}$, find $k$.

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

If $\vec{a}$ is a vector of magnitude 5 and $\vec{b} = 2\hat{i} + \hat{j} - 3\hat{k}$, find the maximum possible value for the scalar projection of $\vec{a}$ on $\vec{b}$.