Test for Collinearity of Vectors

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

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

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

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. 0
3. -2
4. 1

Q. If the vectors $\overrightarrow{\alpha }=a\hat{i}+\hat{j}+\hat{k},\overrightarrow{\beta }=\hat{i}+b\hat{j}+\hat{k}\text{and}\overrightarrow{\gamma }=\hat{i}+\hat{j}+c\hat{k}$ are coplanar where $a\ne 1,b\ne 1,c\ne 1$, then the value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ equals

Q. Let $\overrightarrow{\alpha }=(\lambda -2)\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{\beta }=(4\lambda -2)\overrightarrow{a}+3\overrightarrow{b}$ be two given vectors where vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are non-collinear. The value of $\lambda$ for which vectors $\overrightarrow{\alpha }$ and $\overrightarrow{\beta }$ are collinear, is:

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

Q. The points whose position vectors are $60i+3j,40i-8j$ and $ai-52j$ collinear, if

1. $a=-40$
2. $a=40$
3. $a=20$
4. $a=-20$

Q. Let $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are three unit vectors such that no two of the vectors are collinear. If the vector $\overrightarrow{a}+\overrightarrow{b}$ is collinear with $\overrightarrow{c}$ and the vector $\overrightarrow{b}+\overrightarrow{c}$ is collinear with $\overrightarrow{a},$ then the value of $|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}|$ is

Q. If $a=2\hat{i}-\hat{j}-m\hat{k}$ and $b=\frac{4}{7}\hat{i}-\frac{2}{7}\hat{j}+2\hat{k}$ are collinear, then the value of $m$ is equal to

1. $-7$
2. $-1$
3. $2$
4. $-2$
5. $7$

Q. Assertion :If three points $P,Q$ and $R$ have position vectors $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ respectively, and $2\overrightarrow{a}+3\overrightarrow{b}-5\overrightarrow{c}=0$, then the points $P,Q$ and $R$ must be collinear. Reason: If for three points $A,B$ and $C$; $\overrightarrow{AB}=\lambda \overrightarrow{AC}$, then points $A,B$ and $C$ must be collinear.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect

Q. If $\overline{a}=\hat{i}+\hat{j}-2\hat{k},\overline{b}=2\hat{i}-\hat{j}+\hat{k}$ and $\overline{c}=3\hat{i}-\hat{k}$ then, find the scalars $m$ and $n$ such that $\overline{c}=m\overline{a}+n\overline{b}$.