Scalar Triple Product

Q.

Let $a$, $b$ and $c$ be the distinct non-negative numbers. If the vectors $a\stackrel{\rightharpoonup }{i}+a\stackrel{\rightharpoonup }{j}+c\stackrel{\rightharpoonup }{k}$, $\stackrel{\rightharpoonup }{i}+\stackrel{\rightharpoonup }{k}$and $c\stackrel{\rightharpoonup }{i}+c\stackrel{\rightharpoonup }{j}+b\stackrel{\rightharpoonup }{k}$ lie in a plane, then $c$ is?

1. the harmonic mean of $a\text{and}b$

2. equal to zero

3. the arithmetic mean of $a\text{and}b$

4. the geometric mean of $a\text{and}b$

Q.

What happens when you add a vector to a zero vector?

Q. If $\overrightarrow{p}=\frac{\overrightarrow{b}\times \overrightarrow{c}}{\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]},\overrightarrow{q}=\frac{\overrightarrow{c}\times \overrightarrow{a}}{\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]}$ and $\overrightarrow{r}=\frac{\overrightarrow{a}\times \overrightarrow{b}}{\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]}$, where $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are three non-coplanar vectors, then the value of the expression $(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})\cdot (\overrightarrow{p}+\overrightarrow{q}+\overrightarrow{r})$ is

1. $3$
2. $2$
3. $1$
4. $0$

Q.

Let $\overrightarrow{a}=a_1\hat{i}+{\alpha }_2\hat{j}+a_3\hat{k},\overrightarrow{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}and\overrightarrow{c}=c_1\hat{i}+c_2\hat{j}+c_3\hat{k}$ be three non-zero vectors such that $\overrightarrow{c}$ is a unit vector perpendicular to both the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. If the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\pi }{6},$
then ${\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|}^2$ is equal to

1. 0

2. 1

3. $\frac{1}{4}(a_1^2+a_2^2+a_3^2)(b_1^2+b_2^2+b_3^2)$

4. $\frac{3}{4}(a_1^2+a_2^2+a_3^2)(b_1^2+b_2^2+b_3^2)(c_1^2+c_2^2+c_3^2)$

Q.

If the scalar triple product of 3 vectors $\overrightarrow{a},\overrightarrow{b}and\overrightarrow{c}$is 0 , then they are coplanar.

1. True

2. False

Q. The value of 'a' for which the vectors
A=2i-3j+3k
B=3i+aj-7k
C=5i+3j+6k
are coplanar is

Q.

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$, and $\overrightarrow{d},$ are the unit vectors such that $(\overrightarrow{a}\times \overrightarrow{b}).(\overrightarrow{c}\times \overrightarrow{d})=1$ and $\overrightarrow{a}.\overrightarrow{c}=\frac{1}{2},$ then

1. $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are non-coplanar

2. $\overrightarrow{a},\overrightarrow{b},\overrightarrow{d}$ are non-coplanar

3. $\overrightarrow{b},\overrightarrow{d}$ are non-parallel

4. $\overrightarrow{a},\overrightarrow{d}$ are parallel and
   $\overrightarrow{b},\overrightarrow{c}$ are parallel

Q.

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are non-coplanar vectors and $\overrightarrow{d}=\lambda \overrightarrow{a}+\mu \overrightarrow{b}+\nu \overrightarrow{c},$ then $\lambda$ equal to

1. $\frac{\left[\overrightarrow{d}\overrightarrow{b}\overrightarrow{c}\right]}{\left[\overrightarrow{b}\overrightarrow{a}\overrightarrow{c}\right]}$

2. $\frac{\left[\overrightarrow{b}\overrightarrow{c}\overrightarrow{d}\right]}{\left[\overrightarrow{b}\overrightarrow{c}\overrightarrow{a}\right]}$

3. $\frac{\left[\overrightarrow{b}\overrightarrow{d}\overrightarrow{c}\right]}{\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]}$

4. $\frac{\left[\overrightarrow{c}\overrightarrow{b}\overrightarrow{d}\right]}{\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]}$

Q. If $[\overrightarrow{a}\times \overrightarrow{b}\overrightarrow{b}\times \overrightarrow{c}\overrightarrow{c}\times \overrightarrow{a}]=\lambda {\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]}^2$ then $\lambda$ is equal to

1. $2$
2. $3$
3. $0$
4. $1$

Q. If $\overrightarrow{p}=\frac{\overrightarrow{b}\times \overrightarrow{c}}{\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]},\overrightarrow{q}=\frac{\overrightarrow{c}\times \overrightarrow{a}}{\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]}$ and $\overrightarrow{r}=\frac{\overrightarrow{a}\times \overrightarrow{b}}{\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]}$, where $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are three non-coplanar vectors, then the value of the expression $(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})\cdot (\overrightarrow{p}+\overrightarrow{q}+\overrightarrow{r})$ is

1. $3$
2. $2$
3. $1$
4. $0$

Q.

The volume of parallelopipped formed by following 3 vectors will be___cubic units.

$\overrightarrow{a}=3\hat{i}-2\hat{j}+5\hat{k}\overrightarrow{b}=2\hat{i}+2\hat{j}-\hat{k}\overrightarrow{c}=-4\hat{i}+3\hat{j}+2\hat{k}$

Q. Let $\overrightarrow{a}=\hat{i}-\hat{j},\overrightarrow{b}=\overrightarrow{j}-\hat{k},\overrightarrow{c}=\hat{k}-\hat{i}.If\overrightarrow{d}$ is a unit vector such that $\overrightarrow{a}.\overrightarrow{d}=0=\left[\overrightarrow{b}\overrightarrow{c}\overrightarrow{d}\right],$ then $\overrightarrow{d}$ equals

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

Q.

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$, and $\overrightarrow{d},$ are the unit vectors such that $(\overrightarrow{a}\times \overrightarrow{b}).(\overrightarrow{c}\times \overrightarrow{d})=1$ and $\overrightarrow{a}.\overrightarrow{c}=\frac{1}{2},$ then

1. $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are non-coplanar

2. $\overrightarrow{a},\overrightarrow{b},\overrightarrow{d}$ are non-coplanar

3. $\overrightarrow{b},\overrightarrow{d}$ are non-parallel

4. $\overrightarrow{a},\overrightarrow{d}$ are parallel and
   $\overrightarrow{b},\overrightarrow{c}$ are parallel

Q. Value of scalar triple product is zero if any 2 vectors are identical.

1. True
2. False

Q.

If $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are three non-coplanar vectors, then $(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})\cdot \left[\right(\overrightarrow{a}+\overrightarrow{b})\times (\overrightarrow{a}+\overrightarrow{c}\left)\right]$ equals

1. 0

2. $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]$

3. $2.\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]$

4. $-\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]$

Q.

The volume of tetrahedron whose vertices are

A = (3, 2, 1) , ~B = (1, 2, 4), ~ C = (4, 0, 3), ~ D = (1, 1, 7)~will be $\underset{–}{}$cubic units

1. 5

2. $\frac{5}{6}$

3. $\frac{6}{5}$

4. None of the above