Applications of Dot Product

Q. The vectors $\overrightarrow{x}$ and $\overrightarrow{y}$ satisfy the equation $p\overrightarrow{x}+q\overrightarrow{y}=\overrightarrow{a}$ (where $p,q$ are scalar constants and $\overrightarrow{a}$ is a known vector). It is given that $\overrightarrow{x}.\overrightarrow{y}\ge \frac{|\overrightarrow{a}{|}^2}{4pq}$, then $\frac{|\overrightarrow{x}|}{|\overrightarrow{y}|}$ is equal to ($pq>0$)

1. $1$
2. $\frac{p^2}{q^2}$
3. $\frac{p}{q}$
4. $\frac{q}{p}$

Q.

Which of the following setences are statements? Give reasons for your answer.

(i) There are 35 days in a month.

(ii) Mathematics is difficult.

(iii) The sum of 5 and 7 is greater than 10.

(iv) The square of a number is an even number.

(v) The sides of a quadrilateral have equal length.

(vi) Answer this question.

(vii) The product of (-1) and 8 is 8.

(viii) The sum of all interior angles of a triangle is ${180}^{\circ }$.

(ix) Today is a windy day.

(x) All real numbers are complex numbers.

Q. The component of vector R parallel to X axis is $R\sin \theta$ , where $\theta$ is angle made by the vector with positive direction of X axis.

1. True
2. False

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

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

Q.

The number of complex numbers $z_1$ which can simultaneously satisfy both the equations $|$z - 2$|$ = 2 and z(1 - i) + $\overline{z}(1+i)$ = 4 is equal to

___

Q. Let $\overrightarrow{a}=x\hat{i}+x^2\hat{j}+2\hat{k},\overrightarrow{b}=-3\hat{i}+\hat{j}+\hat{k},\overrightarrow{c}=(3x+11)\hat{i}+(x-9)\hat{j}-3\hat{k}$ be three vectors such that the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is acute and between $\overrightarrow{c}$ and $\overrightarrow{a}$ is obtuse, then x lies between

1. $(-\infty ,1)\cup (2,3)$
   
2. $(-\infty ,0)$
   
3. (4, 5)
4. None of these

Q. Let $\overrightarrow{a}=\hat{i}+\hat{j}+\sqrt{2}\hat{k},\overrightarrow{b}=b_1\hat{i}+b_2\hat{j}+\sqrt{2}\hat{k}$ and $\overrightarrow{c}=5\hat{i}+\hat{j}+\sqrt{2}\hat{k}$ be three vectors such that the projection vector of $\overrightarrow{b}$ on $\overrightarrow{a}$ is $\overrightarrow{a}$. If $\overrightarrow{a}+\overrightarrow{b}$ is perpendicular to $\overrightarrow{c}$, then $|\overrightarrow{b}|$ is equal to :

1. $6$
2. $4$
3. $\sqrt{22}$
4. $\sqrt{32}$

Q. If $\overrightarrow{a}=2\hat{i}-\hat{j}+\hat{k},\overrightarrow{b}=\hat{i}+\hat{j}-2\hat{k}$ and $\overrightarrow{c}=\hat{i}+3\hat{j}-({\lambda }^2+3\lambda )\hat{k}$, where $\lambda$ is a constant and $\overrightarrow{a}$ is perpendicular to $\overrightarrow{c}-\lambda \overrightarrow{b}$, then sum of the different values of $\lambda$ is

1. $-1$
2. $1$
3. $2$
4. $-2$

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

Q.

Find the perpendicular distance of $(1,1)$ from $x-y+\sqrt{2}=0$.

Q. If the vector $\overrightarrow{OP}=\hat{i}+2\hat{j}+2\hat{k}$ rotates through a right angle about origin, passing through the positive $x-$axis on the way becomes $\overrightarrow{OQ}=x\hat{i}+y\hat{j}+z\hat{k}$, then the value of $x-y+z$ is

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

Q. Let $\overrightarrow{b}$ and $\overrightarrow{c}$ be two non-collinear vectors. If $\overrightarrow{a}$ is a vector such that $\overrightarrow{a}\cdot (\overrightarrow{b}+\overrightarrow{c})=4$ and $a\times (\overrightarrow{b}\times \overrightarrow{c})=(x^2-2x+6)\overrightarrow{b}+\left(\sin y\right)\overrightarrow{c},$ then $(x,y)$ lies on the line

1. $x+y=0$
2. $x-y=0$
3. $x=1$
4. $y=\pi$

Q. If $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}$ and $\overrightarrow{r}$ is a non – zero vector such that $p\overrightarrow{r}+(\overrightarrow{r}.\overrightarrow{b})\overrightarrow{a}=\overrightarrow{c}$, then $\overrightarrow{r}$ is equal to

1. $\frac{\overrightarrow{c}}{p}-\frac{(\overrightarrow{b}.\overrightarrow{c})\overrightarrow{a}}{p^2}$
2. $\frac{\overrightarrow{a}}{p}-\frac{(\overrightarrow{c}.\overrightarrow{a})\overrightarrow{b}}{p^2}$
3. $\frac{\overrightarrow{b}}{p}-\frac{(\overrightarrow{a}.\overrightarrow{c})\overrightarrow{c}}{p^2}$
4. $\frac{\overrightarrow{c}}{p^2}-\frac{(\overrightarrow{b}.\overrightarrow{c})\overrightarrow{a}}{p}$

Q. Let $\overrightarrow{a}=2\hat{i}+3\hat{j}-6\hat{k},\overrightarrow{b}=2\hat{i}-3\hat{j}+6\hat{k}$ and $\overrightarrow{c}=-2\hat{i}+3\hat{j}+6\hat{k}$. Let ${\overrightarrow{a}}_1$ be the projection of $\overrightarrow{a}$ on $\overrightarrow{b}$ and ${\overrightarrow{a}}_2$ be the projection of ${\overrightarrow{a}}_1$ on $\overrightarrow{c}$. Then, ${\overrightarrow{a}}_2$ is equal to

1. $\frac{943}{49}(2\hat{i}-3\hat{j}-6\hat{k})$
2. $\frac{943}{{49}^2}(2\hat{i}-3\hat{j}-6\hat{k})$
3. $\frac{943}{49}(-2\hat{i}+3\hat{j}+6\hat{k})$
4. $\frac{943}{{49}^2}(-2\hat{i}+3\hat{j}+6\hat{k})$

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

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

Q. Let $\overrightarrow{a}=2\hat{i}+3\hat{j}-6\hat{k},\overrightarrow{b}=2\hat{i}-3\hat{j}+6\hat{k}$ and $\overrightarrow{c}=-2\hat{i}+3\hat{j}+6\hat{k}$. Let ${\overrightarrow{a}}_1$ be the projection of $\overrightarrow{a}$ on $\overrightarrow{b}$ and ${\overrightarrow{a}}_2$ be the projection of ${\overrightarrow{a}}_1$ on $\overrightarrow{c}$. Then, ${\overrightarrow{a}}_1.\overrightarrow{b}$ is equal to

1. $-41$
2. $-\frac{41}{7}$
3. $41$
4. $287$