Applications Scalar Triple Product

Q. If $\overrightarrow{a}{}^{\prime }=\hat{i}+\hat{j},\overrightarrow{b}{}^{\prime }=\hat{i}+\hat{j}+2\hat{k}$ and $\overrightarrow{c}{}^{\prime }=2\hat{i}+\hat{j}-\hat{k}$. Then altitude of the parallelopiped formed by the vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ having base formed by $\overrightarrow{b}$ and $\overrightarrow{c}$ is $(\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ and ${\overrightarrow{a}}^{\prime },{\overrightarrow{b}}^{\prime },{\overrightarrow{c}}^{\prime }$ are reciprocal system of vectors)

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

Q. Volume of parallelopiped whose coterminous edges are given by $\overrightarrow{u}=\hat{i}+\hat{j}+\lambda \hat{k},\overrightarrow{v}=\hat{i}+\hat{j}+3\hat{k}$ and $\overrightarrow{w}=2\hat{i}+\hat{j}+\hat{k}$ is $1$ cu. unit. If $\theta$ be the angle between the edges $\overrightarrow{u}$ and $\overrightarrow{w}$, then $\cos \theta$ can be:

1. $\frac{7}{6\sqrt{6}}$
2. $\frac{5}{7}$
3. $\frac{7}{6\sqrt{3}}$
4. $\frac{5}{3\sqrt{3}}$

Q. $[b\times cc\times aa\times b]=$

1. [a, b, c]
2. 2 [a, b, c]
3. $[a,b,c{]}^2$
4. \N

Q. Volume of parallelopiped formed by vectors $\overrightarrow{a}\times \overrightarrow{b},\overrightarrow{b}\times \overrightarrow{c}$ and $\overrightarrow{c}\times \overrightarrow{a}$ is $36$ cubic units. Based on the given information above match the following by appropriately matching the lists given in Column I and Column II.

$\begin{array}{cc}Column1& Column2\\ \text{a. Volume of parallelopiped formed by vectors}& \text{p.}0\text{cubic units}\\ \overrightarrow{a},\overrightarrow{b}\text{and}\overrightarrow{c}\text{is}& \\ \text{b. Volume of tetrahedron formed by vectors}& \text{q.}12\text{cubic units}\\ \overrightarrow{a},\overrightarrow{b}\text{and}\overrightarrow{c}\text{is}& \\ \text{c. Volume of parallelopiped formed by vectors}& \text{r.}6\text{cubic units}\\ \overrightarrow{a}+\overrightarrow{b},\overrightarrow{b}+\overrightarrow{c}\text{and}\overrightarrow{c}+\overrightarrow{a}\text{is}& \\ \text{d. Volume of parallelopiped formed by vectors}& \text{s.}1\text{cubic unit}\\ \overrightarrow{a}-\overrightarrow{b},\overrightarrow{b}-\overrightarrow{c}\text{and}\overrightarrow{c}-\overrightarrow{a}\text{is}& \end{array}$

1. $a-r,b-s,c-q,d-p$
2. $a-p,b-s,c-q,d-r$
3. $a-r,b-q,c-s,d-p$
4. $a-s,b-r,c-q,d-p$

Q. If the two diagonals of one of the faces of a parallelopiped are $6\hat{i}+6\hat{k}$ and $4\hat{j}+2\hat{k}$ and one of the edges not containing the given diagonals is $4\hat{j}-8\hat{k}$, then the volume of the parallelopiped is

1. $60$
2. $80$
3. $100$
4. $120$

Q. If a, b, c are linearly independent, then $\frac{[2a+b,2b+c,2c+a]}{[a,b,c]}=$

1. 9
2. 8
3. 7
4. None

Q. Match $\text{List I}$ with the $\text{List II}$ and select the correct answer using the code given below the lists :

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{Area of a triangle with adjacent sides determined by vectors}\overrightarrow{a}\text{and}\overrightarrow{b}\text{is}1\text{. Then the area of}& \text{(P)}& 6\\ & \text{the triangle with adjacent sides determined by}(3\overrightarrow{a}+4\overrightarrow{b})\text{and}(\overrightarrow{a}-3\overrightarrow{b})\text{is}& & \\ \text{(B)}& \text{Volume of parallelopiped determined by vectors}\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\text{is}\frac{1}{4}\text{. Then the volume of the}& \text{(Q)}& 9\\ & \text{parallelopiped determined by vectors}3(\overrightarrow{a}+\overrightarrow{b}),(\overrightarrow{b}+\overrightarrow{c}),4(\overrightarrow{c}+\overrightarrow{a})\text{is}& & \\ \text{(C)}& \text{Area of a parallelogram with adjacent sides determined by vectors}\overrightarrow{a}\text{and}\overrightarrow{b}\text{is}8\text{. Then the}& \text{(R)}& 13\\ & \text{area of the parallelogram with adjacent sides determined by vectors}(2\overrightarrow{a}-\overrightarrow{b})\text{and}\overrightarrow{b}\text{is}& & \\ \text{(D)}& \text{Volume of tetrahedron determined by vectors}\overrightarrow{a},\overrightarrow{b}\text{and}\overrightarrow{c}\text{is}\frac{1}{2}\text{. Then the volume of the}& \text{(S)}& 16\\ & \text{tetrahedron determined by vectors}2(\overrightarrow{a}\times \overrightarrow{b})\text{,}3(\overrightarrow{b}\times \overrightarrow{c})\text{and}(\overrightarrow{c}\times \overrightarrow{a})\text{is}& & \end{array}$

Which of the following is the only CORRECT combination?

1. $\text{(A)}\to \text{(R)},\text{(B)}\to \text{(Q)}$
2. $\text{(A)}\to \text{(R)},\text{(B)}\to \text{(P)}$
3. $\text{(A)}\to \text{(S)},\text{(B)}\to \text{(P)}$
4. $\text{(A)}\to \text{(Q)},\text{(B)}\to \text{(R)}$