Vector Algebra Class 12

Represent graphically a displacement of $40 \mathrm{~km}, 30^{\circ}$ east of north.

Classify the following measures as scalars and vectors.
(i) 10 kg
(ii) 2 meters north-west
(iii) 40°
(iv) 40 watt
(v) $10^{-19}$ coulomb
(vi) $20 \mathrm{~m} / \mathrm{s}^2$

Classify the following as scalar and vector quantities.
(i) time period
(ii) distance
(iii) force
(iv) velocity
(v) work done

In Fig 10.6 (a square), identify the following vectors.
(i) Coinitial
(ii) Equal
(iii) Collinear but not equal

Answer the following as true or false.
(i) $\vec{a}$ and $-\vec{a}$ are collinear.
(ii) Two collinear vectors are always equal in magnitude.
(iii) Two vectors having same magnitude are collinear.
(iv) Two collinear vectors having the same magnitude are equal.

Compute the magnitude of the following vectors:
$$
\vec{a}=\hat{i}+\hat{j}+k ; \quad \vec{b}=2 \hat{i}-7 \hat{j}-3 \hat{k} ; \quad \vec{c}=\frac{1}{\sqrt{3}} \hat{i}+\frac{1}{\sqrt{3}} \hat{j}-\frac{1}{\sqrt{3}} \hat{k}
$$

Write two different vectors having same magnitude.

Write two different vectors having same direction.

Find the values of $x$ and $y$ so that the vectors $2 \hat{i}+3 \hat{j}$ and $x \hat{i}+y \hat{j}$ are equal.

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

Find the sum of the vectors $\vec{a}=\hat{i}-2 \hat{j}+\hat{k}, \vec{b}=-2 \hat{i}+4 \hat{j}+5 \hat{k}$ and $\vec{c}=\hat{i}-6 \hat{j}-7 \hat{k}$.

Find the unit vector in the direction of the vector $\vec{a}=\hat{i}+\hat{j}+2 \hat{k}$.

Find the unit vector in the direction of vector $\overline{\mathrm{PQ}}$, where P and Q are the points (1, 2, 3) and (4, 5, 6), respectively.

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

Find a vector in the direction of vector $5 \hat{i}-\hat{j}+2 \hat{k}$ which has magnitude 8 units.

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 direction cosines of the vector $\hat{i}+2 \hat{j}+3 \hat{k}$.

Find the direction cosines of the vector joining the points $\mathrm{A}(1,2,-3)$ and B (-1, -2, 1), directed from A to B.

Show that the vector $\hat{i}+\hat{j}+\hat{k}$ is equally inclined to the axes $\mathrm{OX}, \mathrm{OY}$ and OZ .

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are $\hat{i}+2 \hat{j}-\hat{k}$ and $-\hat{i}+\hat{j}+\hat{k}$ respectively, in the ratio 2 : 1
(i) internally
(ii) externally

Find the position vector of the mid point of the vector joining the points $\mathrm{P}(2,3,4)$ and $\mathrm{Q}(4,1,-2)$.

Show that the points A, B and C with position vectors, $\vec{a}=3 \hat{i}-4 \hat{j}-4 \hat{k}$, $\vec{b}=2 \hat{i}-\hat{j}+\hat{k}$ and $\vec{c}=\hat{i}-3 \hat{j}-5 \hat{k}$, respectively form the vertices of a right angled triangle.

In triangle ABC (Fig 10.18), which of the following is not true:
(A) $\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{BC}}+\overrightarrow{\mathrm{CA}}=\overrightarrow{0}$
(B) $\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{BC}}-\overrightarrow{\mathrm{AC}}=\overrightarrow{0}$
(C) $\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{BC}}-\overrightarrow{\mathrm{AC}}=\overrightarrow{0}$
(D) $\overrightarrow{\mathrm{AB}}-\overrightarrow{\mathrm{CB}}+\overrightarrow{\mathrm{CA}}=\overrightarrow{0}$

If $\vec{a}$ and $\vec{b}$ are two collinear vectors, then which of the following are incorrect:
(A) $\vec{b}=\lambda \vec{a}$, for some scalar $\lambda$
(B) $\vec{a}= \pm \vec{b}$
(C) the respective components of $\vec{a}$ and $\vec{b}$ are not proportional
(D) both the vectors $\vec{a}$ and $\vec{b}$ have same direction, but different magnitudes.

Find the angle between two vectors $\vec{a}$ and $\vec{b}$ with magnitudes $\sqrt{3}$ and 2 , respectively having $\vec{a} \cdot \vec{b}=\sqrt{6}$.

Find the angle between the vectors $\hat{i}-2 \hat{j}+3 \hat{k}$ and $3 \hat{i}-2 \hat{j}+\hat{k}$

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

Find the projection of the vector $\hat{i}+3 \hat{j}+7 \hat{k}$ on the vector $7 \hat{i}-\hat{j}+8 \hat{k}$.

Show that each of the given three vectors is a unit vector:
$$
\frac{1}{7}(2 \hat{i}+3 \hat{j}+6 \hat{k}), \frac{1}{7}(3 \hat{i}-6 \hat{j}+2 \hat{k}), \quad \frac{1}{7}(6 \hat{i}+2 \hat{j}-3 \hat{k})
$$
Also, show that they are mutually perpendicular to each other.

Find $|\vec{a}|$ and $|\vec{b}|$, if $(\vec{a}+\vec{b}) \cdot(\vec{a}-\vec{b})=8$ and $|\vec{a}|=8|\vec{b}|$.

Evaluate the product $(3 \vec{a}-5 \vec{b}) \cdot(2 \vec{a}+7 \vec{b})$.

Find the magnitude of two vectors $\vec{a}$ and $\vec{b}$, having the same magnitude and such that the angle between them is $60^{\circ}$ and their scalar product is $\frac{1}{2}$.

Find $|\vec{x}|$, if for a unit vector $\vec{a},(\vec{x}-\vec{a}) \cdot(\vec{x}+\vec{a})=12$.

If $\vec{a}=2 \hat{i}+2 \hat{j}+3 \hat{k}, \vec{b}=-\hat{i}+2 \hat{j}+\hat{k}$ and $\vec{c}=3 \hat{i}+\hat{j}$ are such that $\vec{a}+\lambda \vec{b}$ is perpendicular to $\vec{c}$, then find the value of $\lambda$.

Show that $|\vec{a}| \vec{b}+|\vec{b}| \vec{a}$ is perpendicular to $|\vec{a}| \vec{b}-|\vec{b}| \vec{a}$, for any two nonzero vectors $\vec{a}$ and $\vec{b}$.

If $\vec{a} \cdot \vec{a}=0$ and $\vec{a} \cdot \vec{b}=0$, then what can be concluded about the vector $\vec{b}$ ?

If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors such that $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$, find the value of $\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}$.

If either vector $\vec{a}=\overrightarrow{0}$ or $\vec{b}=\overrightarrow{0}$, then $\vec{a} \cdot \vec{b}=0$. But the converse need not be true. Justify your answer with an example.

If the vertices $\mathrm{A}, \mathrm{B}, \mathrm{C}$ of a triangle ABC are $(1,2,3),(-1,0,0),(0,1,2)$, respectively, then find $\angle \mathrm{ABC}$. [ $\angle \mathrm{ABC}$ is the angle between the vectors $\overrightarrow{\mathrm{BA}}$ and $\overline{\mathrm{BC}}$ ].

Show that the points $\mathrm{A}(1,2,7), \mathrm{B}(2,6,3)$ and $\mathrm{C}(3,10,-1)$ are collinear.

Show that the vectors $2 \hat{i}-\hat{j}+\hat{k}, \hat{i}-3 \hat{j}-5 \hat{k}$ and $3 \hat{i}-4 \hat{j}-4 \hat{k}$ form the vertices of a right angled triangle.

If $\vec{a}$ is a nonzero vector of magnitude ' $a$ ' and $\lambda$ a nonzero scalar, then $\lambda \vec{a}$ is unit vector if
(A) $\lambda=1$
(B) $\lambda=-1$
(C) $a=|\lambda|$
(D) $a=1 /|\lambda|$

Find $|\vec{a} \times \vec{b}|$, if $\vec{a}=\hat{i}-7 \hat{j}+7 \hat{k}$ and $\vec{b}=3 \hat{i}-2 \hat{j}+2 \hat{k}$.

Find a unit vector perpendicular to each of the vector $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$, where $\vec{a}=3 \hat{i}+2 \hat{j}+2 \hat{k}$ and $\vec{b}=\hat{i}+2 \hat{j}-2 \hat{k}$.

If a unit vector $\vec{a}$ makes angles $\frac{\pi}{3}$ with $\hat{i}, \frac{\pi}{4}$ with $\hat{j}$ and an acute angle $\theta$ with $\hat{k}$, then find $\theta$ and hence, the components of $\vec{a}$.

Show that
$$
(\vec{a}-\vec{b}) \times(\vec{a}+\vec{b})=2(\vec{a} \times \vec{b})
$$

Find $\lambda$ and $\mu$ if $(2 \hat{i}+6 \hat{j}+27 \hat{k}) \times(\hat{i}+\lambda \hat{j}+\mu \hat{k})=\overrightarrow{0}$.

Given that $\vec{a} \cdot \vec{b}=0$ and $\vec{a} \times \vec{b}=\overrightarrow{0}$. What can you conclude about the vectors $\vec{a}$ and $\vec{b}$ ?

Let the vectors $\vec{a}, \vec{b}, \vec{c}$ be given as $a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}, b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}$, $c_1 \hat{i}+c_2 \hat{j}+c_3 \hat{k}$. Then show that $\vec{a} \times(\vec{b}+\vec{c})=\vec{a} \times \vec{b}+\vec{a} \times \vec{c}$.

If either $\vec{a}=\overrightarrow{0}$ or $\vec{b}=\overrightarrow{0}$, then $\vec{a} \times \vec{b}=\overrightarrow{0}$. Is the converse true? Justify your answer with an example.

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

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

Let the vectors $\vec{a}$ and $\vec{b}$ be such that $|\vec{a}|=3$ and $|\vec{b}|=\frac{\sqrt{2}}{3}$, then $\vec{a} \times \vec{b}$ is a unit vector, if the angle between $\vec{a}$ and $\vec{b}$ is
(A) $\pi / 6$
(B) $\pi / 4$
(C) $\pi / 3$
(D) $\pi / 2$

Area of a rectangle having vertices $\mathrm{A}, \mathrm{B}, \mathrm{C}$ and D with position vectors $-\hat{i}+\frac{1}{2} \hat{j}+4 \hat{k}, \hat{i}+\frac{1}{2} \hat{j}+4 \hat{k}, \hat{i}-\frac{1}{2} \hat{j}+4 \hat{k}$ and $-\hat{i}-\frac{1}{2} \hat{j}+4 \hat{k}$, respectively is
(A) $\frac{1}{2}$
(B) 1
(C) 2
(D) 4

Represent graphically a displacement of $40$ km, $30^\circ$ west of south.

Classify the following measures as scalars and vectors:
(i)
$$
5\text{ seconds}
$$
(ii)
$$
1000\text{ cm}^3
$$
(iii)
$$
10\text{ Newton}
$$
(iv)
$$
30\text{ km/hr}
$$
(v)
$$
10\text{ g/cm}^3
$$
(vi)
$$
20\text{ m/s towards north}
$$

In Fig. 10.5, which of the vectors are:
(i) Collinear
(ii) Equal
(iii) Coinitial

Find the values of $x$, $y$ and $z$ so that the vectors
$$
\vec{a}=x\hat{i}+2\hat{j}+z\hat{k}
$$
and
$$
\vec{b}=2\hat{i}+y\hat{j}+\hat{k}
$$
are equal.

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?

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

Find a vector in the direction of the vector
$$
\vec{a}=\hat{i}-2\hat{j}
$$
that has magnitude $7$ units.

Find the unit vector in the direction of the sum of the vectors
$$
\vec{a}=2\hat{i}+2\hat{j}-5\hat{k}
$$

and

$$
\vec{b}=2\hat{i}+\hat{j}+3\hat{k}.
$$

Write the direction ratios of the vector
$$
\vec{a}=\hat{i}+\hat{j}-2\hat{k}
$$
and hence calculate its direction cosines.

Find the vector joining the points $P(2,3,0)$ and $Q(-1,-2,-4)$, directed from $P$ to $Q$.

Consider two points $P$ and $Q$ with position vectors

$$
\overrightarrow{OP}=3\vec{a}-2\vec{b}
$$

and

$$
\overrightarrow{OQ}=\vec{a}+\vec{b}.
$$

Find the position vector of a point $R$ which divides the line joining $P$ and $Q$ in the ratio $2:1$:
(i) internally

Consider two points $P$ and $Q$ with position vectors

$$
\overrightarrow{OP}=3\vec{a}-2\vec{b}
$$

and

$$
\overrightarrow{OQ}=\vec{a}+\vec{b}.
$$

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

Show that the points
$$
A(2\hat{i}-\hat{j}+\hat{k}),
$$

$$
B(\hat{i}-3\hat{j}-5\hat{k})
$$

and

$$
C(3\hat{i}-4\hat{j}-4\hat{k})
$$
are the vertices of a right-angled triangle.

Find the angle between two vectors $\vec{a}$ and $\vec{b}$ with magnitudes $1$ and $2$, respectively, and when
$$
\vec{a}\cdot\vec{b}=1.
$$

Find the angle $\theta$ between the vectors

$$
\vec{a}=\hat{i}+\hat{j}-\hat{k}
$$

and

$$
\vec{b}=\hat{i}-\hat{j}+\hat{k}.
$$

If
$$
\vec{a}=5\hat{i}-\hat{j}-3\hat{k}
$$

and

$$
\vec{b}=\hat{i}+3\hat{j}-5\hat{k},
$$

then show that the vectors

$$
\vec{a}+\vec{b}
$$

and

$$
\vec{a}-\vec{b}
$$

are perpendicular.

Find the projection of the vector

$$
\vec{a}=2\hat{i}+3\hat{j}+2\hat{k}
$$

on the vector

$$
\vec{b}=\hat{i}+2\hat{j}+\hat{k}.
$$

Find

$$
|\vec{a}-\vec{b}|,
$$

if two vectors $\vec{a}$ and $\vec{b}$ are such that

$$
|\vec{a}|=2,
\qquad
|\vec{b}|=3
$$

and

$$
\vec{a}\cdot\vec{b}=4.
$$

If $\vec{a}$ is a unit vector and

$$
(\vec{x}-\vec{a})\cdot(\vec{x}+\vec{a})=8,
$$

then find

$$
|\vec{x}|.
$$

For any two vectors $\vec{a}$ and $\vec{b}$, we always have

$$
|\vec{a}\cdot\vec{b}|
\leq
|\vec{a}|\,|\vec{b}|.
$$

(Cauchy-Schwarz inequality)

For any two vectors $\vec{a}$ and $\vec{b}$, we always have

$$
|\vec{a}+\vec{b}|
\leq
|\vec{a}|+|\vec{b}|.
$$

(Triangle inequality)

Show that the points

$$
A(-2\hat{i}+3\hat{j}+5\hat{k}),
$$

$$
B(\hat{i}+2\hat{j}+3\hat{k})
$$

and

$$
C(7\hat{i}-\hat{k})
$$

are collinear.

Find

$$
|\vec{a}\times\vec{b}|,
$$

if

$$
\vec{a}=2\hat{i}+\hat{j}+3\hat{k}
$$

and

$$
\vec{b}=3\hat{i}+5\hat{j}-2\hat{k}.
$$

Find a unit vector 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

$$
\vec{b}=\hat{i}+2\hat{j}+3\hat{k}.
$$

Find the area of a triangle having the points

$$
A(1,1,1),\qquad B(1,2,3)
$$

and

$$
C(2,3,1)
$$

as its vertices.

Find the area of a parallelogram whose adjacent sides are given by the vectors

$$
\vec{a}=3\hat{i}+\hat{j}+4\hat{k}
$$

and

$$
\vec{b}=\hat{i}-\hat{j}+\hat{k}.
$$

Write all the unit vectors in the $XY$-plane.

If

$$
\hat{i}+\hat{j}+\hat{k},
$$

$$
2\hat{i}+5\hat{j},
$$

$$
3\hat{i}+2\hat{j}-3\hat{k}
$$

and

$$
\hat{i}-6\hat{j}-\hat{k}
$$

are the position vectors of points $A$, $B$, $C$ and $D$, respectively, then find the angle between

$$
\overrightarrow{AB}
$$

and

$$
\overrightarrow{CD}.
$$

Deduce that $\overrightarrow{AB}$ and $\overrightarrow{CD}$ are collinear.

Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three vectors such that

$$
|\vec{a}|=3,\qquad |\vec{b}|=4,\qquad |\vec{c}|=5,
$$

and each one of them is perpendicular to the sum of the other two. Find

$$
|\vec{a}+\vec{b}+\vec{c}|.
$$

Three vectors $\vec{a}$, $\vec{b}$ and $\vec{c}$ satisfy the condition

$$
\vec{a}+\vec{b}+\vec{c}=\vec{0}.
$$

Evaluate the quantity

$$
\mu
=
\vec{a}\cdot\vec{b}
+\vec{b}\cdot\vec{c}
+\vec{c}\cdot\vec{a},
$$

if

$$
|\vec{a}|=3,\qquad |\vec{b}|=4,\qquad |\vec{c}|=2.
$$

If, with reference to the right-handed system of mutually perpendicular unit vectors $\hat{i}$, $\hat{j}$ and $\hat{k}$,

$$
\vec{\alpha}=3\hat{i}-\hat{j}
$$

and

$$
\vec{\beta}=2\hat{i}+\hat{j}-3\hat{k},
$$

then express $\vec{\beta}$ in the form

$$
\vec{\beta}=\vec{\beta}_1+\vec{\beta}_2,
$$

where $\vec{\beta}_1$ is parallel to $\vec{\alpha}$ and $\vec{\beta}_2$ is perpendicular to $\vec{\alpha}$.

Write down a unit vector in XY-plane, making an angle of $30^{\circ}$ with the positive direction of $x$-axis.

Find the scalar components and magnitude of the vector joining the points $\mathrm{P}\left(x_1, y_1, z_1\right)$ and $\mathrm{Q}\left(x_2, y_2, z_2\right)$.

A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of north and stops. Determine the girl's displacement from her initial point of departure.

If $\vec{a}=\vec{b}+\vec{c}$, then is it true that $|\vec{a}|=|\vec{b}|+|\vec{c}|$ ? Justify your answer.

Find the value of $x$ for which $x(\hat{i}+\hat{j}+\hat{k})$ is a unit vector.

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

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

Show that the points $\mathrm{A}(1,-2,-8), \mathrm{B}(5,0,-2)$ and $\mathrm{C}(11,3,7)$ are collinear, and find the ratio in which B divides AC.

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are $(2 \vec{a}+\vec{b})$ and $(\vec{a}-3 \vec{b})$ externally in the ratio 1 : 2. Also, show that P is the mid point of the line segment RQ.

The two adjacent sides of a parallelogram are $2 \hat{i}-4 \hat{j}+5 \hat{k}$ and $\hat{i}-2 \hat{j}-3 \hat{k}$. Find the unit vector parallel to its diagonal. Also, find its area.

Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are $\pm\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)$.

Let $\vec{a}=\hat{i}+4 \hat{j}+2 \hat{k}, \vec{b}=3 \hat{i}-2 \hat{j}+7 \hat{k}$ 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 $\vec{c} \cdot \vec{d}=15$.

The scalar product of the vector $\hat{i}+\hat{j}+\hat{k}$ with a 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 one. Find the value of $\lambda$.

If $\vec{a}, \vec{b}, \overrightarrow{\mathrm{c}}$ are mutually perpendicular vectors of equal magnitudes, show that the vector $\vec{c} \cdot \vec{d}=15$ is equally inclined to $\vec{a}, \vec{b}$ and $\vec{c}$.

Prove that $(\vec{a}+\vec{b}) \cdot(\vec{a}+\vec{b})=|\vec{a}|^2+|\vec{b}|^2$, if and only if $\vec{a}, \vec{b}$ are perpendicular, given $\vec{a} \neq \overrightarrow{0}, \vec{b} \neq \overrightarrow{0}$.

If $\theta$ is the angle between two vectors $\vec{a}$ and $\vec{b}$, then $\vec{a} \cdot \vec{b} \geq 0$ only when
(A) $0<\theta<\frac{\pi}{2}$
(B) $0 \leq \theta \leq \frac{\pi}{2}$
(C) $0<\theta<\pi$
(D) $0 \leq \theta \leq \pi$

Let $\vec{a}$ and $\vec{b}$ be two unit vectors and $\theta$ is the angle between them. Then $\vec{a}+\vec{b}$ is a unit vector if
(A) $\theta=\frac{\pi}{4}$
(B) $\theta=\frac{\pi}{3}$
(C) $\theta=\frac{\pi}{2}$
(D) $\theta=\frac{2 \pi}{3}$

The value of $\hat{i} \cdot(\hat{j} \times \hat{k})+\hat{j} \cdot(\hat{i} \times \hat{k})+\hat{k} \cdot(\hat{i} \times \hat{j})$ is
(A) 0
(B) -1
(C) 1
(D) 3

If $\theta$ is the angle between any two vectors $\vec{a}$ and $\vec{b}$, then $|\vec{a} \cdot \vec{b}|=|\vec{a} \times \vec{b}|$ when $\theta$ is equal to
(A) 0
(B) $\frac{\pi}{4}$
(C) $\frac{\pi}{2}$
(D) $\pi$