Vector Algebra

A scalar is a quantity that is determined by its:
1. Direction
2. Magnitude
3. Both magnitude and direction
4. None of these
A vector is a quantity that is determined by its:
1. Magnitude
2. Direction
3. Both magnitude and direction
4. None of these
If we have a liberty to choose the origin of the vector at any point then it is said to be:
1. Free vector
2. Localised vector
3. Support of a vector
4. Terminal point
Co-initial vectors are having the same:
1. Terminal point
2. Initial point
3. Both terminal and initial points
4. None of these
If two or more vectors lie on the same plane or parallel to the same plane, they are called as:
1. Collinear
2. Co-initial
3. Co-terminal
4. Coplanar
Two vectors are said to be equal if they have:
1. Same direction and same magnitude
2. Same magnitude only
3. Same direction only
4. None of these
$\vec{0}$ is known as:
1. Zero vector
2. Null vector
3. Void vector
4. All options are correct
Vector addition is:
1. Asociative
2. Commutative
3. Both associative and commutative
4. None of these
Let $\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}$ be the position vector of any point and let $α, β, γ$ be the directed angles of $\vec{r}$ . Then $\sin^{2} (α) + \sin^{2} (β) + \sin^{2} (γ)$ is:
1. 1
2. 3
3. 0
4. 2
Let $\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}$ be the position vector of any point and let $α, β, γ$ be the direction angles of $\vec{r}$ . Then the sum of squares of the direction cosines of $\vec{r}$ is:
1. 1
2. 2
3. 3
4. 4
Let G be the centroid of $△ A B C$ . If $\vec{A B} = \vec{a}$ , $\vec{A C} = \vec{b}$ , then the bisector $A G$ , in terms of $\vec{a}$ and $\vec{b}$ is:
1. $\frac{2}{3} (\vec{a} + \vec{b})$
2. $\frac{1}{6} (\vec{a} + \vec{b})$
3. $\frac{1}{3} (\vec{a} + \vec{b})$
4. $\frac{1}{2} (\vec{a} + \vec{b})$
If ABCDEF is a regular hexagon, then $\vec{A B} + \vec{E B} + \vec{F C}$ equals:
1. $2 \vec{A B}$
2. $\vec{0}$
3. $3 \vec{A B}$
4. $4 \vec{A B}$
If $\vec{a}$ and $\vec{b}$ are unit vectors, then which of the following values of $\vec{a} . \vec{b}$ is not possible?
1. $\sqrt{3}$
2. $\frac{\sqrt{3}}{2}$
3. $\frac{1}{\sqrt{2}}$
4. $\frac{- 1}{2}$
The vector component of $\vec{b}$ perpendiculat to $\vec{a}$ is:
1. $(\vec{b} . \vec{c}) \vec{a}$
2. $\frac{\vec{a} \times (\vec{b} \times \vec{a})}{{|\begin{matrix} \vec{a} \end{matrix}|}^{2}}$
3. $\vec{a} \times (\vec{b} \times \vec{a})$
4. None of these
If $\vec{a}$ is any vector, then ${(\vec{a} \times \hat{i})}^{2} + {(\vec{b} \times \hat{j})}^{2} + {(\vec{c} \times \hat{k})}^{2} =$ :
1. ${\vec{a}}^{2}$
2. $2 {\vec{a}}^{2}$
3. $3 {\vec{a}}^{2}$
4. $4 {\vec{a}}^{2}$
The vector $\vec{b} = 3 \hat{i} + 4 \hat{k}$ is to be written as the sum of a vector $\vec{α}$ parallel to $\vec{a} = \hat{i} + \hat{j}$ and a vector $\vec{β}$ perpendicular to $\vec{a}$ . Then $\vec{α}$ is:
1. $\frac{3}{2} (\hat{i} + \hat{j})$
2. $\frac{2}{3} (\hat{i} + \hat{j})$
3. $\frac{1}{2} (\hat{i} + \hat{j})$
4. $\frac{1}{3} (\hat{i} + \hat{j})$
The value of $[\vec{a} - \vec{b}, \vec{b} - \vec{c}, \vec{c} - \vec{a}]$ , where $|\vec{a}| = 1$ , $|\vec{b}| = 5$ , $|\vec{c}| = 3$ is:
1. 0
2. 1
3. 6
4. None of these
If $\vec{r} . \vec{a} = \vec{r} . \vec{b} = \vec{r} . \vec{c} = 0$ for some non-zero vector $\vec{r}$ , then the value of $[\vec{a} \vec{b} \vec{c}]$ is:
1. 2
2. 3
3. 0
4. None of these
For every point P (x, y, z) on the xy-plane:
1. x = 0
2. y = 0
3. z = 0
4. x = y = z = 0
A rectangular parallelopiped is formed by planes drawn through the points (5, 7, 9) and (2, 3, 7) parallel to the coordinate planes. The length of an edge of this rectangular parallelopiped is:
1. 2
2. 3
3. 4
4. All of these
What is the direction cosines of Y-axis?
1. 1, 0, 0
2. 0, 1, 0
3. 0, 0, 1
4. None of these
Ratio in which the xy-plane divides the join of (1, 2, 3) and (4, 2, 1) is:
1. 3 : 1 internally
2. 3 : 1 externally
3. 1 : 2 internally
4. 1 : 2 externally
If $[2 \vec{a} + 4 \vec{b} \vec{c} \vec{d}] = λ [\vec{a} \vec{c} \vec{d}] + μ [\vec{b} \vec{c} \vec{d}]$ , then $λ + μ$ is:
1. 6
2. -6
3. 10
4. 8
If the vectors $4 \hat{i} + 11 \hat{j} + m \hat{k}$ , $7 \hat{i} + 2 \hat{j} + 6 \hat{k}$ and $\hat{i} + 5 \hat{j} + 4 \hat{k}$ are coplanar, then m is:
1. 0
2. 38
3. -10
4. 10
A unit vector perpendicular to both $\hat{i} + \hat{j}$ and $\hat{j} + \hat{k}$ is:
1. $\hat{i} - \hat{j} + \hat{k}$
2. $\hat{i} + \hat{j} + \hat{k}$
3. $\frac{1}{\sqrt{3}} (\hat{i} + \hat{j} + \hat{k})$
4. $\frac{1}{\sqrt{3}} (\hat{i} - \hat{j} + \hat{k})$
If $\hat{i}, \hat{j}, \hat{k}$ are unit vectors, then:
1. $\hat{i} . \hat{j} = 1$
2. $\hat{i} . \hat{i} = 1$
3. $\hat{i} \times \hat{j} = 1$
4. $\hat{i} \times (\hat{j} \times \hat{k}) = 1$
The projection of the vector $\hat{i} + \hat{j} + \hat{k}$ along the vector of $\hat{j}$ is:
1. 1
2. 0
3. 2
4. -1
If the vectors $3 \hat{i} + λ \hat{j} + \hat{k}$ and $2 \hat{i} - \hat{j} + 8 \hat{k}$ are perpendicular, then $λ$ is equal to:
1. -14
2. 7
3. 14
4. -7
ABCD is a parallelogram with AC and BD as diagonals. Then $\vec{A C} - \vec{B D}$ is:
1. $4 \vec{A B}$
2. $3 \vec{A B}$
3. $2 \vec{A B}$
4. $A B$
If points $A (60 \vec{i} + 3 \vec{j})$ , $B (40 \vec{i} - 8 \vec{j})$ and $C (a \vec{i} - 52 \vec{j})$ are collinear, then a is equal to:
1. 40
2. -40
3. 20
4. -20

Vector Algebra - I

Quiz