Applications of Dot Product

Q. Let u, v, w be such that |u|=1, |v|=2, |w|=3. If the projection v along u is equal to that of w along u and v, w are perpendicular to each other then |u-v+w| =

1. 2
2. $\sqrt{14}$
3. $\sqrt{7}$
4. 14

Q. If a+b+c=0 and |a|=3, |b|=4 and $|c|=\sqrt{37}$, the angle between a and b is

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

Q. If $f^2\left(x\right)+2g^2\left(x\right)+3h^2\left(x\right)\le 1$ and $u\left(x\right)=2f\left(x\right)-2g\left(x\right)+3h\left(x\right),$ then the maximum value of $u^2\left(x\right)$ is equal to

Q. If, in a right angled triangle ABC, the hypotenuse AB = p, then AB . AC + BC . BA + CA . CB =

1. $2p^2$
2. $\frac{p^2}{2}$
3. $p^2$
4. $None$

Q. The length of the projection of $2\hat{i}-3\hat{j}+\hat{k}$ in the direction of $4\hat{i}-4\hat{j}+7\hat{k}$ is

1. 3
2. 9
3. 27
4. $3\sqrt{3}$

Q. If $a=4\hat{i}+6\hat{j}$ and $b=3\hat{j}+4\hat{k}$, then the vector form of the component of a along b is

1. $\frac{18(3\hat{j}+4\hat{k})}{10\sqrt{3}}$
2. $\frac{18(3\hat{j}+4\hat{k})}{25}$
3. $\frac{18(3\hat{j}+4\hat{k})}{\sqrt{13}}$
4. $3\hat{j}+\hat{k}$

Q. If |a|=3, |b|=4 and |a+b|=1, then |a-b|=

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

Q. If $a=-\hat{i}+2\hat{j}+3\hat{k}$ and $b=\hat{i}-2\hat{j}+\hat{k}$, $c=4\hat{i}-3\hat{j}-2\hat{k}$ and $a+\lambda b$ is perpendicular to c, then $\lambda$ =

1. 1
2. 3
3. 2
4. 4

Q. Let a and b be two unit vectors. If the vectors c=a+2b and d=5a-4b are perpendicular to each other, then the angle between a and b is

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

Q. A particle acted on by constant forces $4\hat{i}+\hat{j}-3\hat{k}$ and $3\hat{i}+\hat{j}-\hat{k}$ is displaced from the point $\hat{i}+2\hat{j}+3\hat{k}$ to the point $5\hat{i}+4\hat{j}+\hat{k}$. The total work done by the forces is

1. 20 units
2. 30 units
3. 40 units
4. 50 units