Fundamental Theorem In 3D

Q.

Classify the following as scalar and vector quantities:
(i) Velocity

Q. Classify the given measure as scalar and vector:
(i) ${10}^{-19}C$

Q. If the angle between $\overrightarrow{a}=2x^2\hat{i}+4x\hat{j}+\hat{k}$ and $\overrightarrow{b}=7\hat{i}-2\hat{j}+x\hat{k}$ is obtuse and the angle between $\overrightarrow{b}$ and $Z-$ axis is acute and less than $\frac{\pi }{6},$ then

1. $x\in \varphi$
2. $x\in (0,\frac{1}{2})$
3. $x\in (-\sqrt{159},\sqrt{159})$
4. $x\in (-1,1)$

Q.

Classify the following as scalar and vector quantities:
(i) Distance

Q. What do you mean by scalar and vector quantity??

Q.

Classify the following as scalar and vector quantities:
(i) Force

Q.

If the direction cosines of a line are k, k and k, then

(a) k>0 (b)0<k<1

(c) k=1 (d)$k=\frac{1}{\sqrt{3}}or-\frac{1}{\sqrt{3}}$

Q. For the vectors $\overline{x}=(1,2,1);\overline{y}=(2,-3,-1)$ Component of $\overline{y}$ along $\overline{x}=$ _____

1. $-\frac{5}{\sqrt{14}}$
2. $\frac{5}{14}$
3. $-\frac{5}{14}$
4. $\frac{5}{\sqrt{14}}$

Q.

$If\overrightarrow{a},\overrightarrow{b}\overrightarrow{c}areunitvectorssuchthat\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0},thenthevalueof\overrightarrow{a}.\overrightarrow{b}+\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{c}.\overrightarrow{a}is$

(a) 1 (b) 3

(c) $\frac{-3}{2}$ (d) None of these

Q. Classify the following as scalar and vector quantities:

Q. If $a+b+c=0$ and $|a|=3;|b|=5;|c|=7$, find the angle between a vector and b

1. ${60}^0$
2. ${30}^0$
3. ${45}^0$
4. $noneofthese$

Q. Classify the following as scalar and vector quantities:

Q. Classify the following measure as scalar and vector:
$15$ kg

Q. Classify the following measure as scalar and vector:
$20$ kg weight

Q. Classify the following as scalar and vector quantities:

Q. Classify the following measure as scalar and vector:
${45}^{\circ }$

Q. Find the direction cosines of the vector joining the point $A(1,2,-3)$ and $B(-1,-2,1)$ , directed from A and B.

Q. If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are unit vectors such that $\overrightarrow{a}\cdot \overrightarrow{b}=0$, $(\overrightarrow{a}-\overrightarrow{c})\cdot (\overrightarrow{b}+\overrightarrow{c})=0$ and $\overrightarrow{c}=\lambda \overrightarrow{a}+\mu \overrightarrow{b}+\omega (\overrightarrow{a}\times \overrightarrow{b})$, where $\lambda ,\mu ,\omega$ are scalars then?

1. ${\mu }^2+{\omega }^2=1$
2. $\lambda -\mu =1$
3. $(\mu +1{)}^2+{\mu }^2+{\omega }^2=1$
4. ${\lambda }^2+{\mu }^2=1$

Q. The vector component of the vector $\hat{i}+\hat{j}+\hat{k}$ along the vector $2\hat{i}-\hat{j}+2\hat{k}$ is

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

Q. If $\overrightarrow{a}=p\overrightarrow{i}+3\overrightarrow{j}-7\overrightarrow{k},\overrightarrow{b}=p\overrightarrow{i}-p\overrightarrow{j}+4\overrightarrow{k}$ and if the angle between $a$ and $b$ is acute, then the values of $p$ lies in

1. $p<-4$ or $p>7$
2. $(-7,4)$
3. $p\le -4$ and $p\ge 7$
4. $[-7,4]$