Applications of Cross Product

Q. Let $\overrightarrow{A}$ be vector parallel to line of intersection of planes $P_1$ and $P_2$ through origin. $P_1$ is parallel to the vectors $2\hat{j}+3\hat{k}$ and $4\hat{j}-3\hat{k}$ and $P_2$ is parallel to $\hat{j}-\hat{k}$ and $3\hat{i}+3\hat{j},$ then the angle between vector $\overrightarrow{A}$ and $2\overrightarrow{i}+\overrightarrow{j}-2\hat{k}$ is

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

Q. If $\overrightarrow{a}$ and $\overrightarrow{b}$ are the two sides of a triangle, then the area of triangle will be given by $|\overrightarrow{a}\times \overrightarrow{b}|$

1. True
2. False

Q. A unit vector making an obtuse angle with x – axis and perpendicular to the plane containing the points $\hat{i}-2\hat{j}+3\hat{k},2\hat{i}-3\hat{j}+4\hat{k}$ and $\hat{i}-5\hat{j}+7\hat{k}$,

1. Also makes and obtuse angle with y - axis
2. Also makes an obtuse angle with z - axis
3. Also makes an obtuse angle with y and z axis
4. None of the above

Q. The area enclosed by the points (1, 1), (-1, 1), (-1, -1) and (1, -1) is

Q.

If $\overrightarrow{r}\times \overrightarrow{b}=\overrightarrow{c}\times \overrightarrow{b}$ and $\overrightarrow{r}.\overrightarrow{a}=0$ where $\overrightarrow{a}=2\hat{i}+3\hat{j}-\hat{k},\overrightarrow{b}=3\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{c}=\hat{i}+\hat{j}+3\hat{k}$, then $\overrightarrow{r}$ is equal to

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

2. $2(\hat{i}+\hat{j}+\hat{k})$

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

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

Q. The value of b such that the scalar product of the vector $\hat{i}+\hat{j}+\hat{k}$ with the unit vector parallel to the sum of the vectors $2\hat{i}+4\hat{j}+5\hat{k}$ and $b\hat{i}+2\hat{j}+3\hat{k}$ is one, is

1. -1
2. 1
3. 0
4. -2

Q. If a is any vector then $(a\times \hat{i}{)}^2+(a\times \hat{j}{)}^2+(a\times \hat{k}{)}^2$ =

1. $3a^2$
2. $4a^2$
3. $a^2$
4. $2a^2$

Q. If $\overrightarrow{AB}=3\hat{i}-2\hat{j}+\hat{k}$, $\overrightarrow{BC}=\hat{i}+2\hat{j}-\hat{k}$, $\overrightarrow{CD}=2\hat{i}+\hat{j}+3\hat{k}$, $\overrightarrow{OA}=\hat{i}+\hat{j}+\hat{k}$, then the position vector of $\overrightarrow{OD}$ is

1. $7\hat{i}-2\hat{j}-3\hat{k}$
2. $7\hat{i}-2\hat{j}+3\hat{k}$
3. $7\hat{i}+2\hat{j}-3\hat{k}$
4. $7\hat{i}+2\hat{j}+4\hat{k}$

Q. The area of the parallelogram whose diagonals are $\hat{i}-3\hat{j}+2\hat{k},-\hat{i}+2\hat{j}$ is

1. $4\sqrt{29}sq$.units
2. $\frac{1}{2}\sqrt{21}sq$.units
3. $10\sqrt{3}sq$.units
4. $\frac{1}{2}\sqrt{270}sq$.units

Q. If $2\hat{i}+3\hat{j}+4\hat{k}$ and $\hat{i}-\hat{j}+\hat{k}$ are two adjacent sides of a parallelogram, then the area of the parallelogram will be

1. 
2. 
3. Cannot be calculated from the given data
4. none of the above

Q. The vector area of the triangle with vertices $\hat{i}+\hat{j}+\hat{k},\hat{i}+\hat{j}+2\hat{k},\hat{i}+2\hat{j}+\hat{k}$ is

1. $\hat{i}$
2. $-\hat{i}$
3. $\frac{1}{2}i$
4. $-\frac{1}{2}i$

Q. If the vectors $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ form the sides BC, CA and AB respectively of a $\Delta ABC$, then

1. $\overrightarrow{a}.\overrightarrow{b}=\overrightarrow{b}.\overrightarrow{c}=\overrightarrow{c}.\overrightarrow{a}$
2. $\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{c}\times \overrightarrow{a}=\overrightarrow{0}$
3. $\overrightarrow{a}.\overrightarrow{b}+\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{c}.\overrightarrow{a}=0$
4. $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{c}\times \overrightarrow{a}$

Q. The area of a triangle is 5. Two of its vertices are (2, 1) and (3, -2). The third vertex lies on y=x+3. Find the third vertex.

Q. If $G$ is the centroid of $\Delta ABC$ and $O$ is any point then $\overline{OA}+\overline{OB}+\overline{OC}=$

1. $\overline{OG}$
2. $2\overline{OG}$
3. $3\overline{OG}$
4. $4\overline{OG}$