Find the unit vector in the direction of sum of vectors $\overrightarrow{\mathbf{a}}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\overrightarrow{\mathbf{b}}=2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$.
If $\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and $\overrightarrow{\mathbf{b}}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$, then find the unit vector in the direction of
(i) $6 \overrightarrow{\mathbf{b}}\quad$ (ii) $2 \overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}$
Find a unit vector in the direction of $\overrightarrow{\mathbf{P Q}}$, where $P$ and $Q$ have coordinates $(5,0,8)$ and $(3,3,2)$, respectively.
If $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$ are the position vectors of $\overrightarrow{\mathbf{A}}$ and $\overrightarrow{\mathbf{B}}$ respectively, then find the position vector of a point $\overrightarrow{\mathbf{C}}$ in $\overrightarrow{\mathbf{B A}}$ produced such that $\overrightarrow{\mathbf{B C}}=1.5 \overrightarrow{\mathbf{B A}}$.
Using vectors, find the value of $k$, such that the points $(k,-10,3)$, $(1,-1,3)$ and $(3,5,3)$ are collinear.
A vector $\overrightarrow{\mathbf{r}}$ is inclined at equal angles to the three axes. If the magnitude of $\overrightarrow{\mathbf{r}}$ is $2 \sqrt{3}$ units, then find the value of $\overrightarrow{\mathbf{r}}$.
If a vector $\overrightarrow{\mathbf{r}}$ has magnitude 14 and direction ratios 2,3 and -6 . Then, find the direction cosines and components of $\overrightarrow{\mathbf{r}}$, given that $\overrightarrow{\mathbf{r}}$ makes an acute angle with $X$-axis.
Find a vector of magnitude 6 , which is perpendicular to both the vectors $$ 2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}} \text { and } 4 \hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}} .$$
Find the angle between the vectors $2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-\hat{\mathbf{k}}$.
If $\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}=0$, then show that $\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}=\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}$. Interpret the result geometrically.
Find the sine of the angle between the vectors $\overrightarrow{\mathbf{a}}=3 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and $\overrightarrow{\mathbf{b}}=2 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$.
If $A, B, C$ and $D$ are the points with position vectors $\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$, $2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \quad 2 \hat{\mathbf{i}}-3 \hat{\mathbf{k}}$ and $3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ respectively, then find the projection of $\overrightarrow{\mathbf{A B}}$ along $\overrightarrow{\mathbf{C D}}$.
Using vectors, find the area of the $\triangle A B C$ with vertices $A(1,2,3)$, $B(2,-1,4)$ and $C(4,5,-1)$.
Using vectors, prove that the parallelogram on the same base and between the same parallels are equal in area.
Prove that in any $\triangle A B C, \cos A=\frac{b^2+c^2-a^2}{2 b c}$, where $a, b$ and $c$ are the magnitudes of the sides opposite to the vertices $A, B$ and $C$, respectively.
If $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ determine the vertices of a triangle, show that $\frac{1}{2}[\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}]$ gives the vector area of the triangle. Hence, deduce the condition that the three points $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ are collinear. Also, find the unit vector normal to the plane of the triangle.
Show that area of the parallelogram whose diagonals are given by $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$ is $\frac{|\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}|}{2}$. Also, find the area of the parallelogram, whose diagonals are $2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+k$ and $\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-\hat{\mathbf{k}}$.
If $\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\overrightarrow{\mathbf{b}}=\hat{\mathbf{j}}-\hat{\mathbf{k}}$, then find a vector $\overrightarrow{\mathbf{c}}$ such that $\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{c}}=3$.
The vector in the direction of the vector $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ that has magnitude 9 is
The position vector of the point which divides the join of points $2 \overrightarrow{\mathbf{a}}-3 \overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}$ in the ratio $3: 1$, is
The vector having initial and terminal points as $(2,5,0)$ and $(-3,7$, 4), respectively is
The angle between two vectors $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$ with magnitudes $\sqrt{3}$ and 4, respectively and $\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=2 \sqrt{3}$ is
Find the value of $\lambda$ such that the vectors $\overrightarrow{\mathbf{a}}=2 \hat{\mathbf{i}}+\lambda \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ are orthogonal.
The value of $\lambda$ for which the vectors $3 \hat{\mathbf{i}}-6 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $2 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+\lambda \hat{\mathbf{k}}$ are parallel, is
The vectors from origin to the points $A$ and $B$ are $\overrightarrow{\mathbf{a}}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and $\overrightarrow{\mathbf{b}}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ respectively, then the area of $\triangle O A B$ is equal to
For any vector $\overrightarrow{\mathbf{a}}$, the value of $(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}})^2+(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}})^2+(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}})^2$ is
If $|\overrightarrow{\mathbf{a}}|=10,|\overrightarrow{\mathbf{b}}|=2$ and $\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=12$, then the value of $|\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}|$ is
The vectors $\lambda \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}, \hat{\mathbf{i}}+\lambda \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\lambda \hat{\mathbf{k}}$ are coplanar, if
If $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ are unit vectors such that $\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}=\overrightarrow{0}$, then the value of $\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{b}} \cdot \overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{c}} \cdot \overrightarrow{\mathbf{a}}$ is
The projection vector of $\overrightarrow{\mathbf{a}}$ on $\overrightarrow{\mathbf{b}}$ is
If $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ are three vectors such that $\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}=\overrightarrow{0}$ and $|\overrightarrow{\mathbf{a}}|=2$, $|\overrightarrow{\mathbf{b}}|=3$ and $|\overrightarrow{\mathbf{c}}|=5$, then the value of $\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{b}} \cdot \overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{c}} \cdot \overrightarrow{\mathbf{a}}$ is
If $|\overrightarrow{\mathbf{a}}|=4$ and $-3 \leq \lambda \leq 2$, then the range of $|\lambda \overrightarrow{\mathbf{a}}|$ is
The number of vectors of unit length perpendicular to the vectors $\overrightarrow{\mathbf{a}}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and $\overrightarrow{\mathbf{b}}=\hat{\mathbf{j}}+\hat{\mathbf{k}}$ is
The vector $\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}$ bisects the angle between the non-collinear vectors $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$, if...... .
If $\overrightarrow{\mathbf{r}} \cdot \overrightarrow{\mathbf{a}}=0, \overrightarrow{\mathbf{r}} \cdot \overrightarrow{\mathbf{b}}=0$ and $\overrightarrow{\mathbf{r}} \cdot \overrightarrow{\mathbf{c}}=0$ for some non-zero vector $\overrightarrow{\mathbf{r}}$, then the value of $\overrightarrow{\mathbf{a}} \cdot(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})$ is ......... .
The vectors $\overrightarrow{\mathbf{a}}=3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and $\overrightarrow{\mathbf{b}}=-\hat{\mathbf{i}}-2 \hat{\mathbf{k}}$ are the adjacent sides of a parallelogram. The angle between its diagonals is ............. .
The values of $k$, for which $|k \overrightarrow{\mathbf{a}}|<\overrightarrow{\mathbf{a}} \mid$ and $k \overrightarrow{\mathbf{a}}+\frac{1}{2} \overrightarrow{\mathbf{a}}$ is parallel to $\overrightarrow{\mathbf{a}}$ holds true are ............. .
The value of the expression $|\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}|^2+(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}})^2$ is .................. .
If $|\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}|^2+|\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}|^2=144$ and $|\overrightarrow{\mathbf{a}}|=4$, then $|\overrightarrow{\mathbf{b}}|$ is equal to ......... .
If $\overrightarrow{\mathbf{a}}$ is any non-zero vector, then $(\overrightarrow{\mathbf{a}} \cdot \hat{\mathbf{i}}) \cdot \hat{\mathbf{i}}+(\overrightarrow{\mathbf{a}} \cdot \hat{\mathbf{j}}) \cdot \hat{\mathbf{j}}+(\overrightarrow{\mathbf{a}} \cdot \hat{\mathbf{k}}) \hat{\mathbf{k}}$ is equal to ........... .
If $|\overrightarrow{\mathbf{a}}|=|\overrightarrow{\mathbf{b}}|$, then necessarily it implies $\overrightarrow{\mathbf{a}}= \pm \overrightarrow{\mathbf{b}}$.
Position vector of a point $\overrightarrow{\mathbf{P}}$ is a vector whose initial point is origin.
If $|\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}|=|\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}|$, then the vectors $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$ are orthogonal
The formula $(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}})^2=\overrightarrow{\mathbf{a}}^2+\overrightarrow{\mathbf{b}}^2+2 \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}$ is valid for non-zero vectors $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$.
If $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$ are adjacent sides of $\mathbf{a}$ rhombus, then $\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=0$