Applications of Cross Product

Q.

Let $\overrightarrow{a}=2\overrightarrow{i}+\overrightarrow{j}-2\overrightarrow{k}$ and $\overrightarrow{b}=\overrightarrow{i}+\overrightarrow{j}$. Let $c$ be a vector such that $\left|c-a\right|=3$, $\left|\left(a\times b\right)\times c\right|=3$ and the angle between $a\times b$ and $c$ is $30°$. Then $a.c$ is equal to

1. $2$

2. $5$

3. $\frac{1}{8}$

4. $\frac{25}{8}$

Q.

Evaluate ${\sum }_{k=1}^{11}(2+3^k)$

Q. In a $△ABC,$ $P$ be an interior point such that $\overrightarrow{PA}+2\overrightarrow{PB}+3\overrightarrow{PC}=0.$ The ratio of the area of $△ABC$ to that of $△APC$ is

1. $3:1$
2. $3:2$
3. $5:2$
4. $2:1$

Q.

If vertices of any quadrilateral are (0, -1), (2, 1), (0, 3) and (- 2, 1), then it is a

1. Square

2. Collinear

3. Parallelogram

4. Rectangle

Q.

The base of a triangle is the line x+y=1.One of the vertices is (3, 4).So the area of the triangle is:

1. 3 units

2. 6 units

3. 2√2 units

4. 3√2 units

Q. If $\overrightarrow{a}$ and $\overrightarrow{b}$ are the two sides of a triangle, then the area of triangle will be given by $|\overrightarrow{a}\times \overrightarrow{b}|$

1. True
2. False

Q. If $2\hat{i}+3\hat{j}+4\hat{k}$ and $\hat{i}-\hat{j}+\hat{k}$ are two adjacent sides of a parallelogram, then the area of the parallelogram will be

1. $\sqrt{87}$
2. $\sqrt{70}$
3. Cannot be calculated from the given data
4. none of the above

Q.

$\frac{sin5\theta +sin2\theta -sin\theta }{cos5\theta +2cos3\theta +2co{s}^2\theta +cos\theta }$ is equal to

1. 

2. 

3. 

4. None of these

Q. A force of 500 N is applied to a wrench as shown in the figure. The force is being applied at an angle of ${235}^{\circ }$ with positive x – axis

The length of the wrench is 0.3 m The magnitude of vector moment of the force F acting at A about origin is given by _____ and the screw at origin will move in ____ direction.

1. $500\times 0.3\times sin{235}^{\circ }$ negative z direction
2. $500\times 0.3\times sin{55}^{\circ }$ positive z direction
3. $500\times 0.3\times sin{125}^{\circ }$ negative z direction
4. $500\times 0.3\times sin{135}^{\circ }$positive z direction

Q. Vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ satisfying the vector equation $\overrightarrow{A}+\overrightarrow{B}=\overrightarrow{a},\overrightarrow{A}\times \overrightarrow{B}=\overrightarrow{b}$ and $\overrightarrow{A}\cdot \overrightarrow{a}=1$, where $\overrightarrow{a}$ and $\overrightarrow{b}$ are given vectors, are

1. $\overrightarrow{A}=\frac{(\overrightarrow{a}\times \overrightarrow{b})-\overrightarrow{a}}{|\overrightarrow{a}{|}^2}$
2. $\overrightarrow{B}=\frac{(\overrightarrow{b}\times \overrightarrow{a})+\overrightarrow{a}(|\overrightarrow{a}{|}^2-1)}{|\overrightarrow{a}{|}^2}$
3. $\overrightarrow{A}=\frac{(\overrightarrow{a}\times \overrightarrow{b})+\overrightarrow{a}}{|\overrightarrow{a}{|}^2}$
4. $\overrightarrow{B}=\frac{(\overrightarrow{b}\times \overrightarrow{a})-\overrightarrow{a}(|\overrightarrow{a}{|}^2-1)}{|\overrightarrow{a}{|}^2}$

Q. Find the area of the region {(x, y) : y²<=4x , 4x²+4y²<=9

Q. If $z=5x+y$ subject to the constraints $3x+y\le 15,4x+3y\le 30$ where $x,y\ge 0,$ then the value of $z_{max}$ is equal to

Q. Let $\overrightarrow{A}$ be vector parallel to line of intersection of planes $P_1$ and $P_2$ through origin. $P_1$ is parallel to the vectors $2\hat{j}+3\hat{k}$ and $4\hat{j}-3\hat{k}$ and $P_2$ is parallel to $\hat{j}-\hat{k}$ and $3\hat{i}+3\hat{j},$ then the angle between vector $\overrightarrow{A}$ and $2\overrightarrow{i}+\overrightarrow{j}-2\hat{k}$ is

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

Q. A unit vector making an obtuse angle with x – axis and perpendicular to the plane containing the points $\hat{i}-2\hat{j}+3\hat{k},2\hat{i}-3\hat{j}+4\hat{k}$ and $\hat{i}-5\hat{j}+7\hat{k}$,

1. Also makes and obtuse angle with y - axis
2. Also makes an obtuse angle with z - axis
3. Also makes an obtuse angle with y and z axis
4. None of the above

Q. If $\overrightarrow{a}$ and $\overrightarrow{b}$ are unequal vectors such that $(\overrightarrow{a}-\overrightarrow{b})\times \left[\right(\overrightarrow{b}+\overrightarrow{a})\times (2\overrightarrow{a}+\overrightarrow{b}\left)\right]=\overrightarrow{a}+\overrightarrow{b}$, then the angle $\theta$ between $\overrightarrow{a}$ and $\overrightarrow{b}$ is

1. $0$
2. $\frac{\pi }{2}$
3. $\frac{\pi }{4}$
4. $\pi$