Cross Product of Two Vectors

Q. Let $△PQR$ be a triangle. Let $\overrightarrow{a}=\overrightarrow{QR},\overrightarrow{b}=\overrightarrow{RP}$ and $\overrightarrow{c}=\overrightarrow{PQ}.$ If $|\overrightarrow{a}|=12,|\overrightarrow{b}|=4\sqrt{3}$ and $\overrightarrow{b}.\overrightarrow{c}=24,$ then which of the following is (are) true?

1. $\frac{|\overrightarrow{c}{|}^2}{2}-|\overrightarrow{a}|=12$
2. $\frac{|\overrightarrow{c}{|}^2}{2}+|\overrightarrow{a}|=30$
3. $|\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{c}\times \overrightarrow{a}|=48\sqrt{3}$
4. $\overrightarrow{a}.\overrightarrow{b}=-72$

Q.

Let the vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ such that $\left|\overrightarrow{a}\right|=2$, $\left|\overrightarrow{b}\right|=4$ and $\left|\overrightarrow{c}\right|=4$. If the projection of vector $b$ on vector $a$ is equal to the projection of vector $c$ on vector $a$ and $b$ is perpendicular to vector $c$, then the value of $\left|\overrightarrow{a}+\overrightarrow{b}-\overrightarrow{c}\right|$ is :

Q. If $\theta$ is the angle between two vectors $\hat{i}-2\hat{j}+3\hat{k}$ and $3\hat{i}-2\hat{j}+\hat{k}$, find $\sin \theta$.

Q. The perpendicular distance of the point $P(6,7,8)$ from $XY-$plane is

1. $8$
2. $7$
3. $6$
4. $5$

Q. Let $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ be three vectors such that $|\overrightarrow{a}|=\sqrt{3},|\overrightarrow{b}|=5,\overrightarrow{b}.\overrightarrow{c}=10$ and the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ is $\frac{\pi }{3}.$
If $\overrightarrow{a}$ is perpendicular to vector $\overrightarrow{b}\times \overrightarrow{c}$, then $|\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})|$ is equal to

Q. Let $\hat{u}=u_1\hat{i}+u_2\hat{j}+u_3\hat{k}$ be a unit vector in $R^3$ and $\hat{w}=\frac{1}{\sqrt{6}}(\hat{i}+\hat{j}+2\hat{k}).$ Given that there exists a vector $\overrightarrow{v}$ in $R^3$ such that $|\hat{u}\times \overrightarrow{v}|=1$ and $\hat{w}\cdot (\hat{u}\times \overrightarrow{v})=1.$ Which of the following statement(s) is(are) correct ?

1. There is exactly one choice for such $\overrightarrow{v}$
2. There are infinitely many choices for such $\overrightarrow{v}$
3. If $\hat{u}$ lies in the $xy$-plane then $|u_1|=|u_2|$
4. If $\hat{u}$ lies in the $xz$-plane then $2|u_1|=|u_3|$

Q. Given that P=Q=R. If P vector +Q vector =R vector then the angle between P vector and R vector is A. If (P +Q +R) vector =0vector then the angle between P vector and R vector is A 2. Then fin the relation between A1and A1?

Q.

If $A,B$and $C$ are three non-coplanar vectors, then $(A+B+C).\left[\right(A+B)\times (A+C\left)\right]$ equals?

1. $0$

2. $[A,B,C]$

3. $2[A,B,C]$

4. $-[A,B,C]$

Q.

If $\alpha ,\beta$ are natural numbers such that ${100}^{\alpha }–199\beta =\left(100\right)\left(100\right)+\left(99\right)\left(101\right)+\left(98\right)\left(102\right)+\dots .+\left(1\right)\left(199\right)$, then the slope of the line passing through $(\alpha ,\beta )$ and origin is

1. $510$

2. $550$

3. $540$

4. $530$

Q.

The minimum value of $\left|x\right|+\left|x+\frac{1}{2}\right|+\left|x-3\right|+\left|x-\frac{5}{2}\right|$ is

1. $0$

2. $2$

3. $4$

4. $6$

Q.

Given that $a.b=0$ and $a\times b=0$. What can you conclude about the vectors a and b?

Q. A unit vector perpendicular to the plane containing the vector $\hat{i}+2\hat{j}+\hat{k}$ and $-2\hat{i}+\hat{j}+3\hat{k}$ is

1. $\frac{-\hat{i}+\hat{j}-\hat{k}}{\sqrt{3}}$
2. $\frac{-\hat{i}+\hat{j}-\hat{k}}{\sqrt{3}}$
3. $\frac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}}$
4. $\frac{-\hat{i}-\hat{j}-\hat{k}}{\sqrt{3}}$

Q. If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are non coplanar non zero vectors such that $\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{a},\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c}$ and $\overrightarrow{c}\times \overrightarrow{a}=\overrightarrow{b},$ then which of the following is not correct?

1. $|\overrightarrow{a}|=1$
2. $\overrightarrow{a}\cdot (\overrightarrow{b}\times \overrightarrow{c})=1$
3. $|\overrightarrow{a}|\ne |\overrightarrow{b}|\ne |\overrightarrow{c}|$
4. $|\overrightarrow{a}|+|\overrightarrow{b}|+|\overrightarrow{c}|=3$

Q. A unit vector perpendicular to the plane determined by the points P(1, -1, 2), Q(2, 0, -1) and R(0, 2, 1) is

1. $\frac{2\hat{i}-\hat{j}-\hat{k}}{3}$
2. $\frac{2\hat{i}+\hat{j}+\hat{k}}{3}$
3. $\frac{2\hat{i}+\hat{j}+\hat{k}}{\sqrt{6}}$
4. $\frac{2\hat{i}-\hat{j}-\hat{k}}{\sqrt{3}}$

Q. Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5).

Q. A unit vector perpendicular to the plane of $a=2\hat{i}-6\hat{j}-3\hat{k},b=4\hat{i}+3\hat{j}-\hat{k}$ is

1. $\frac{4\hat{i}+3\hat{j}-\hat{k}}{\sqrt{26}}$
2. $\frac{2\hat{i}-6\hat{j}-3\hat{k}}{7}$
3. $\frac{3\hat{i}-2\hat{j}+6\hat{k}}{7}$
4. $\frac{2\hat{i}-3\hat{j}-6\hat{k}}{7}$

Q.

The vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are not perpendicular and $\overrightarrow{c}$ and $\overrightarrow{d}$ are two vectors satisfying$\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{b}\times \overrightarrow{d}$ and $\overrightarrow{a}·\overrightarrow{d}=0$. Then, the vector $\overrightarrow{d}$ is equal to?

1. $\overrightarrow{c}+\left(\frac{\overrightarrow{a}·\overrightarrow{c}}{\overrightarrow{a}·\overrightarrow{d}}\right)\overrightarrow{b}$

2. $\overrightarrow{b}+\left(\frac{\overrightarrow{b}·\overrightarrow{c}}{\overrightarrow{a}·\overrightarrow{b}}\right)\overrightarrow{c}$

3. $\overrightarrow{c}-\left(\frac{\overrightarrow{a}·\overrightarrow{c}}{\overrightarrow{a}·\overrightarrow{b}}\right)\overrightarrow{b}$

4. $\overrightarrow{b}-\left(\frac{\overrightarrow{b}·\overrightarrow{c}}{\overrightarrow{a}·\overrightarrow{b}}\right)\overrightarrow{c}$

Q.

The shoppers who come to a departmental store are marked as:$\mathrm{man}\left(\mathrm{M}\right)$ , $\mathrm{woman}\left(\mathrm{W}\right)$ , $\mathrm{boy}\left(\mathrm{B}\right)\mathrm{or}\mathrm{girl}\left(\mathrm{G}\right)$ . The following list gives the shoppers who came during the first hour in the morning. $$\begin{gathered} \mathrm{W}\mathrm{W}\mathrm{W}\mathrm{G}\mathrm{B}\mathrm{W}\mathrm{W}\mathrm{M}\mathrm{G}\mathrm{G}\mathrm{M}\mathrm{M}\mathrm{W}\mathrm{W}\mathrm{W}\mathrm{W}\mathrm{G}\mathrm{B}\mathrm{M}\mathrm{W}\mathrm{B}\mathrm{G}\mathrm{G}\mathrm{M}\mathrm{W}\mathrm{W}\mathrm{M}\mathrm{M}\mathrm{W}\mathrm{W}\mathrm{W}\mathrm{M}\mathrm{W}\mathrm{B}\mathrm{W}\mathrm{G}\mathrm{M}\mathrm{W}\mathrm{W}\mathrm{W}\mathrm{W}\mathrm{G}\mathrm{W}\mathrm{M}\mathrm{M}\mathrm{W}\mathrm{M}\mathrm{W} \\ \mathrm{G}\mathrm{W}\mathrm{M}\mathrm{G}\mathrm{W}\mathrm{M}\mathrm{M}\mathrm{B}\mathrm{G}\mathrm{G}\mathrm{W}. \end{gathered}$$ Make a frequency distribution table using tally marks. Draw a bar graph to illustrate it.

Q. The projection of the vector $\hat{i}+\hat{j}+\hat{k}$ on the line whose vector equation is $\overrightarrow{r}=(3+t)\hat{i}+(2t-1)\hat{j}+3t\hat{k},t$ being the scalar parameter is

1. $\frac{6}{\sqrt{14}}$
2. $\frac{1}{\sqrt{14}}$
3. $10$
4. $6$

Q. The distance between the parallel lines $\overrightarrow{r}=2\hat{i}+3\hat{j}-\hat{k}+\lambda (\hat{i}-\hat{j}+2\hat{k})$ and $\overrightarrow{r}=-3\hat{i}+4\hat{j}+\hat{k}+\mu (\hat{i}-\hat{j}+2\hat{k})$ is

1. $2\sqrt{\frac{22}{3}}$
2. $\sqrt{\frac{80}{3}}$
3. $\frac{14}{\sqrt{6}}$
4. $7$

Q. Let $\hat{\alpha },\hat{\beta },\hat{\gamma }$ be three unit vectors such that $\hat{\alpha }\times (\hat{\beta }\times \hat{\gamma })=\frac{1}{2}(\hat{\beta }+\hat{\gamma })$ where $\hat{\alpha }\times (\hat{\beta }\times \hat{\gamma })=(\hat{\alpha }.\hat{\gamma })\hat{\beta }-(\hat{\alpha }.\hat{\beta })\hat{\gamma }$. If $\hat{\beta }$ is not parallel to $\hat{\gamma }$, then the angle between $\hat{\alpha }$ and $\hat{\beta }$ is

1. $\frac{5\pi }{6}$
2. $\frac{\pi }{6}$
3. $\frac{2\pi }{3}$
4. $\frac{\pi }{3}$

Q. If $(\overrightarrow{a},\overrightarrow{b})=\frac{\pi }{6},\overrightarrow{c}$ is perpendicular to $\overrightarrow{a}$and$\overrightarrow{b},\left|\overrightarrow{a}\right|=3,\left|\overrightarrow{b}\right|=4,\left|\overrightarrow{c}\right|=6,$$\text{then}\left|\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]\right|$ is equal to

1. $12\sqrt{3}$
2. $48\sqrt{3}$
3. $36$
4. $76$

Q. $\overrightarrow{a}$ and $\overrightarrow{b}$ are two vectors such that $|\overrightarrow{a}|=1,|\overrightarrow{b}|=4$ and $\overrightarrow{a}\cdot \overrightarrow{b}=2$. If $\overrightarrow{c}=(2\overrightarrow{a}\times \overrightarrow{b})-3\overrightarrow{b}$, then the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ is

1. $\frac{3\pi }{4}$
2. $\frac{5\pi }{6}$
3. $\frac{\pi }{3}$
4. $\frac{\pi }{6}$

Q.

For the non - zero vectors $a,b$and $c$ the relation$a.\left(b\times c\right)=0$ is true, if?

1. $b\perp c$

2. $a\perp b$

3. $a\left|\right|c$

4. $a\perp c$

Q.

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are three non-zero vectors, no two of which are collinear, $\overrightarrow{a}+\overrightarrow{b}$ is collinear with $\overrightarrow{c}$ and $\overrightarrow{b}+3\overrightarrow{c}$ is collinear with $\overrightarrow{a}$, then $|\overrightarrow{a}+2\overrightarrow{b}+6\overrightarrow{c}|$ will be equal to

1. zero

2. 1

3. 9

4. None of these

Q. Find a unit vector perpendicular to each of the vector and , where and .

Q. The area of the figure formed by the points (-1, -1, 1);(1, 1, 1) and their mirror images on the plane 3x + 2y + 4z + 1 = 0 is

1. $\frac{5\sqrt{33}}{29}$
2. $\frac{4\sqrt{33}}{29}$
3. $\frac{20\sqrt{33}}{27}$
4. $\frac{20\sqrt{33}}{29}$

Q. Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two vectors of equal magnitude $5$ uints. Let $\overrightarrow{p},\overrightarrow{q}$ be vectors such that $\overrightarrow{p}=\overrightarrow{a}-\overrightarrow{b}$ and $\overrightarrow{q}=\overrightarrow{a}+\overrightarrow{b}$. If $|\overrightarrow{p}\times \overrightarrow{q}|=2\{\gamma -(\overrightarrow{a}.\overrightarrow{b}{)}^2{\}}^{1/2}$, then the value of $\gamma$ is

1. $5$
2. $25$
3. $125$
4. $625$

Q. Let $P$ be an interior point of $\Delta ABC$ such that $4\overrightarrow{PA}+3\overrightarrow{PB}+5\overrightarrow{PC}=0$. If $\text{Area}\left(\Delta ABC\right)=k\times \text{Area}\left(\Delta APC\right)$, then $k$ is

Q. If the moduli of the vectors a, b, c are 3, 4, 5 respectively and a and b + c, b and c + a, c and a + b are mutually perpendicular, then the modulus of a + b + c is

1. √12
2. 5√2
3. 12
4. 50