Orthogonal System of Vectors

Q. Let $\overrightarrow{a}=2\hat{i}-\hat{j}+2\hat{k}$ and $\overrightarrow{b}=\hat{i}+2\hat{j}-\hat{k}$. Let a vector $\overrightarrow{v}$ be in the plane containing $\overrightarrow{a}$ and $\overrightarrow{b}$. If $\overrightarrow{v}$ is perpendicular to the vector $3\hat{i}+2\hat{j}-\hat{k}$ and its projection on $\overrightarrow{a}$ is $19$ units, then $|2\overrightarrow{v}{|}^2$ is equal to

Q. If $\hat{i},\hat{j}$ and $\hat{k}$ are the unit vectors along positive direction of x, y and z axes respectively and $\overrightarrow{r}=a\hat{i}+b\hat{j}+c\hat{k}$ then the component of $\overrightarrow{r}$ along z axis is b.

1. True
2. False

Q. (i) Find a unit vector perpendicular to both the vectors $4\hat{i}-\hat{j}+3\hat{k}\mathrm{and}-2\hat{i}+\hat{j}-2\hat{k}.$

(ii) Find a unit vector perpendicular to the plane containing the vectors $\overrightarrow{a}=2\hat{i}+\hat{j}+\hat{k}\mathrm{and}\overrightarrow{b}=\hat{i}+2\hat{j}+\hat{k}.$

Q. In a four dimensional space, where unit vectors along the axes are $\hat{i},\hat{j},\hat{k}$ and $\hat{l},$$\overrightarrow{a_1},\overrightarrow{a_2},\overrightarrow{a_3},\overrightarrow{a_4}$ are four non-zero vectors such that no vector can be expressed as a linear combination of others and $(\lambda -1)({\overrightarrow{a}}_1-{\overrightarrow{a}}_2)+\alpha ({\overrightarrow{a}}_2+{\overrightarrow{a}}_3)+\gamma ({\overrightarrow{a}}_3+{\overrightarrow{a}}_4-2{\overrightarrow{a}}_2)+{\overrightarrow{a}}_3+\delta {\overrightarrow{a}}_4=\overrightarrow{0}.$
Then which of the following is/are correct?

1. $\lambda =1$
2. $\alpha =2$
3. $\gamma =1$
4. $\delta =\frac{1}{3}$

Q. Find the unit vector in the direction of $3\hat{i}+4\hat{j}-12\hat{k}.$

Q. Find a unit vector parallel to the vector $\hat{i}+\sqrt{3}\hat{j}$.

Q. Find a unit vector in the direction of the resultant of the vectors $\hat{i}-\hat{j}+3\hat{k},2\hat{i}+\hat{j}-2\hat{k}\mathrm{and}\hat{i}+2\hat{j}-2\hat{k}.$

Q. Let $\overrightarrow{p},\overrightarrow{q},\overrightarrow{r}$ be three mutually perpendicular vectors of the same magnitude. If a vector $\overrightarrow{x}$ satisfies the equation
$\overrightarrow{p}\left[\right(\overrightarrow{x}-\overrightarrow{q})\times \overrightarrow{p}]+\overrightarrow{q}\times \left[\right(\overrightarrow{x}-\overrightarrow{r})\times \overrightarrow{q}]+\overrightarrow{r}\times \left[\right(\overrightarrow{x}-\overrightarrow{p})\times \overrightarrow{r}]=\overrightarrow{0}$, then $\overrightarrow{x}$ is given by

1. $\frac{1}{2}(\overrightarrow{p}+\overrightarrow{q}-2\overrightarrow{r})$
2. $\frac{1}{2}(\overrightarrow{p}+\overrightarrow{q}+\overrightarrow{r})$
3. $\frac{1}{3}(\overrightarrow{p}+\overrightarrow{q}+\overrightarrow{r})$
4. $\frac{1}{3}(2\overrightarrow{p}+\overrightarrow{q}-\overrightarrow{r})$

Q. Find a vector of magnitude of 5 units parallel to the resultant of the vectors $\overrightarrow{a}=2\hat{i}+3\hat{j}-\hat{k}\mathrm{and}\overrightarrow{b}=\hat{i}-2\hat{j}+\widehat{k.}$

Q. Write a unit vector in the direction of $\overrightarrow{a}=3\hat{i}+2\hat{j}+6\hat{k}.$

Q. If $\overrightarrow{a}=3\hat{i}-\hat{j}+2\hat{k}$ and $\overrightarrow{b}=2\hat{i}+\hat{j}-\hat{k},$ then find $\left(\overrightarrow{a}\times \overrightarrow{b}\right)\overrightarrow{a}.$

Q. $\int \frac{2x-1}{(x-1)(x+2)(x-3)}dx=A\log |x-1|+B\log |x+2|+C\log |x-3|+K$. Then $A,B,C$ are respectively

1. $\frac{1}{6},\frac{-1}{3},\frac{1}{3}$
2. $\frac{1}{6},\frac{1}{3},\frac{1}{5}$
3. $\frac{-1}{6},\frac{1}{3},\frac{1}{3}$
4. $\frac{-1}{6},\frac{-1}{3},\frac{1}{2}$

Q. If $\overrightarrow{a}=\hat{i}+\hat{j}-\hat{k},\overrightarrow{b}=-\hat{i}+2\hat{j}+2\hat{k}\mathrm{and}\overrightarrow{c}=-\hat{i}+2\hat{j}-\hat{k},$ then a unit vector normal to the vectors $\overrightarrow{a}+\overrightarrow{b}\mathrm{and}\overrightarrow{b}-\overrightarrow{c}$ is
(a) $\hat{i}$

(b) $\hat{j}$

(c) $\hat{k}$

(d) none of these

Q. If $\hat{i},\hat{j},\hat{k}$are unit vectors, then
(a) $\hat{i}·\hat{j}=1$

(b) $\hat{i}·\hat{i}=1$

(c) $\hat{i}\times \hat{j}=1$

(d) $\hat{i}\times \left(\hat{j}\times \hat{k}\right)=1$

Q. If the vectors $3\hat{i}+m\hat{j}+\hat{k}\mathrm{and}2\hat{i}-\hat{j}-8\hat{k}$ are orthogonal, find m.

Q. If $\overrightarrow{a}$is a non-zero vector of magnitude 'a' and λ is a non-zero scalar, then λ$\overrightarrow{a}$ is a unit vector if
(a) λ = 1
(b) λ = −1
(c) a = |λ|
(d) $a=\frac{1}{\left|\lambda \right|}$

Q. The unit vector in the direction of sum of the vectors $\overrightarrow{A}=2\hat{i}+4\hat{j}-7\hat{k}$ and $\overrightarrow{B}=4\hat{i}-6\hat{j}+4\hat{k}$ will be

1. $6\hat{i}-2\hat{j}-3\hat{k}$
2. $\hat{i}-\hat{j}-\hat{k}$
3. $\frac{6\hat{i}-2\hat{j}-3\hat{k}}{7}$

Q. A unit vector perpendicular to both $\hat{i}+\hat{j}\mathrm{and}\hat{j}+\hat{k}$ is

(a) $\hat{i}-\hat{j}+\hat{k}$

(b) $\hat{i}+\hat{j}+\hat{k}$

(c) $\frac{1}{\sqrt{3}}\left(\hat{i}+\hat{j}+\hat{k}\right)$

(d) $\frac{1}{\sqrt{3}}\left(\hat{i}-\hat{j}+\hat{k}\right)$

Q. If $\overrightarrow{a}=x\hat{i}+2\hat{j}-z\hat{k}\mathrm{and}\overrightarrow{b}=3\hat{i}-y\hat{j}+\hat{k}$ are two equal vectors, then write the value of x + y + z.

Q. Let $\overrightarrow{p},\overrightarrow{q},\overrightarrow{r}$ be three mutually perpendicular vectors of the same magnitude. If a vector $\overrightarrow{x}$ satisfies the equation
$\overrightarrow{p}\left[\right(\overrightarrow{x}-\overrightarrow{q})\times \overrightarrow{p}]+\overrightarrow{q}\times \left[\right(\overrightarrow{x}-\overrightarrow{r})\times \overrightarrow{q}]+\overrightarrow{r}\times \left[\right(\overrightarrow{x}-\overrightarrow{p})\times \overrightarrow{r}]=\overrightarrow{0}$, then $\overrightarrow{x}$ is given by

1. $\frac{1}{3}(2\overrightarrow{p}+\overrightarrow{q}-\overrightarrow{r})$
2. $\frac{1}{2}(\overrightarrow{p}+\overrightarrow{q}-2\overrightarrow{r})$
3. $\frac{1}{2}(\overrightarrow{p}+\overrightarrow{q}+\overrightarrow{r})$
4. $\frac{1}{3}(\overrightarrow{p}+\overrightarrow{q}+\overrightarrow{r})$

Q. If $\hat{i},\hat{j}$ and $\hat{k}$ are the unit vectors along positive direction of x, y and z axes respectively and $\overrightarrow{r}=a\hat{i}+b\hat{j}+c\hat{k}$ then the component of $\overrightarrow{r}$ along z axis is b.

1. True
2. False

Q. If $\overrightarrow{a}=\hat{i}+2\hat{j}-3\hat{k}\mathrm{and}\overrightarrow{b}=2\hat{i}+4\hat{j}+9\hat{k},$ find a unit vector parallel to $\overrightarrow{a}+\overrightarrow{b}$.

Q. Find a unit vector in the direction of $\overrightarrow{a}=2\hat{i}-3\hat{j}+6\hat{k}$.

Q. Let $\overrightarrow{p},\overrightarrow{q},\overrightarrow{r}$ be three mutually perpendicular vectors of the same magnitude. If a vector $\overrightarrow{x}$ satisfies the equation
$\overrightarrow{p}\left[\right(\overrightarrow{x}-\overrightarrow{q})\times \overrightarrow{p}]+\overrightarrow{q}\times \left[\right(\overrightarrow{x}-\overrightarrow{r})\times \overrightarrow{q}]+\overrightarrow{r}\times \left[\right(\overrightarrow{x}-\overrightarrow{p})\times \overrightarrow{r}]=\overrightarrow{0}$, then $\overrightarrow{x}$ is given by

1. $\frac{1}{2}(\overrightarrow{p}+\overrightarrow{q}-2\overrightarrow{r})$
2. $\frac{1}{2}(\overrightarrow{p}+\overrightarrow{q}+\overrightarrow{r})$
3. $\frac{1}{3}(\overrightarrow{p}+\overrightarrow{q}+\overrightarrow{r})$
4. $\frac{1}{3}(2\overrightarrow{p}+\overrightarrow{q}-\overrightarrow{r})$

Q. Find a unit vector in the direction of the vector $\overrightarrow{a}=3\hat{i}-2\hat{j}+6\hat{k}$.

Q. If $\hat{i},\hat{j}$ and $\hat{k}$ are the unit vectors along positive direction of x, y and z axes respectively and $\overrightarrow{r}=a\hat{i}+b\hat{j}+c\hat{k}$ then the component of $\overrightarrow{r}$ along z axis is b.

1. True
2. False

Q. Write a unit vector in the direction of the sum of the vectors $\overrightarrow{a}=2\hat{i}+2\hat{j}-5\hat{k}$ and $\overrightarrow{b}=2\hat{i}+\hat{j}-7\hat{k}$. [CBSE 2014]

Q. Find the value of x for which $x\left(\hat{i}+\hat{j}+\hat{k}\right)$ is a unit vector.

Q.

determine a unit vector perpendicular to both vector a=2i+k and vector b=i-j+2k

Q. Find all vectors of magnitude $10\sqrt{3}$ that are perpendicular to the plane of $\hat{i}+2\hat{j}+\hat{k}$ and $-\hat{i}+3\hat{j}+4\hat{k}$. [NCERT EXEMPLAR]