Solve this following
Question: If $(\vec{a}+\vec{b}) \cdot(\vec{a}-\vec{b})=8$ and $|\vec{a}|=8|\vec{b}|$, find $|\vec{a}|$ and $|\vec{b}|$. Solution:...
Read More →Solve this following
Question: If $|\vec{a}|=2,|\vec{b}|=3$ and $\vec{a} \cdot \vec{b}=4$, find $|\vec{a}-\vec{b}|$. Solution:...
Read More →Find the angle between
Question: Find the angle between $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$, when i. $|\overrightarrow{\mathrm{a}}|=2,|\overrightarrow{\mathrm{b}}|=1$ and $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=\sqrt{3}$ ii. $|\overrightarrow{\mathrm{a}}|=|\overrightarrow{\mathrm{b}}|=\sqrt{2}$ and $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=-1$ Solution:...
Read More →Solve this following
Question: If $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0},|\vec{a}|=3,|\vec{b}|=5$ and $|\vec{c}|=7$, find the angle between $\vec{a}$ and $\vec{b}$. Solution:...
Read More →Solve this following
Question: Prove that $(\vec{a}+\vec{b}) \cdot(\vec{a}-\vec{b})=|\vec{a}|^{2}+|\vec{b}|^{2} \Leftrightarrow \vec{a} \perp \vec{b}$, where $\vec{a} \neq \overrightarrow{0}$ and $\vec{b} \neq \overrightarrow{0}$. Solution:...
Read More →Solve this following
Question: Express the vector $\overrightarrow{\mathrm{a}}=(6 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}-6 \hat{\mathrm{k}})$ as sum of two vectors such that one is parallel to the vector $\overrightarrow{\mathrm{b}}=(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})$ and the other is perpendicular to $\overrightarrow{\mathrm{b}}$. Solution:...
Read More →Find the angle between
Question: Find the angle between $(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}})$ and $(\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}})$, if $\overrightarrow{\mathrm{a}}=(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+3 \hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{b}}=(3 \hat{\mathrm{i}}+\hat{\mathrm{j}}+2 \hat{\mathrm{k}})$. Solution:...
Read More →Solve this following
Question: Find a vector $\vec{a}$ of magnitude $5 \sqrt{2}$, making an angle $\pi / 4$ with $x$ - axis, $\pi / 2$ with $y$ - axis and an acute angle $\theta$ with z - axis. Solution:...
Read More →Solve this following
Question: Show that the vector $\overrightarrow{\mathrm{a}}=(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})$ is equally inclined to the coordinate axes. Solution:...
Read More →Find the angles which the vector
Question: Find the angles which the vector $\vec{a}=3 \hat{i}-6 \hat{j}+2 \hat{k}$ makes with the coordinate axes. Solution:...
Read More →Solve this following
Question: If $\vec{a}$ is a unit vector such that $(\vec{x}-\vec{a}) \cdot(\vec{x}+\vec{a})=8$, find $|\vec{x}|$. Solution:...
Read More →Solve this following
Question: If $\overrightarrow{\mathrm{a}}=(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{b}}=(3 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})$ then calculate the angle between $(2 \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}})$ and $(\overrightarrow{\mathrm{a}}+2 \overrightarrow{\mathrm{b}})$. Solution:...
Read More →Find the angle between the vectors
Question: Find the angle between the vectors $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$, when i. $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ ii. $\overrightarrow{\mathrm{a}}=3 \hat{\mathrm{i}}+\hat{\mathrm{j}}+2 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}$ iii. $\overrightarrow{\mathr...
Read More →Solve this following
Question: i. Find the projection of $\overrightarrow{\mathrm{a}}$ on $\overrightarrow{\mathrm{b}}$ if $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=8$ and $\overrightarrow{\mathrm{b}}=(2 \hat{\mathrm{i}}+6 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$. ii. Write the projection of the vector $(\hat{\mathrm{i}}+\hat{\mathrm{j}})$ on the vector $(\hat{\mathrm{i}}-\hat{\mathrm{j}})$. Solution: 0...
Read More →Write the projection of vector
Question: Write the projection of vector $(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})$ along the vector $\hat{\mathrm{j}}$. Solution:...
Read More →Find the projection of
Question: Find the projection of $(8 \hat{\mathrm{i}}+\hat{\mathrm{j}})$ in the direction of $(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}})$. Solution:...
Read More →Solve this following
Question: Let $\overrightarrow{\mathrm{a}}=(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{b}}=(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}})$. Find the projection of (i) $\overrightarrow{\mathrm{a}}$ on $\overrightarrow{\mathrm{b}}$ and (ii) $\overrightarrow{\mathrm{b}}$ on $\overrightarrow{\mathrm{a}}$. Solution:...
Read More →Solve this following
Question: Let $\overrightarrow{\mathrm{a}}=4 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}-\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}-4 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}=3 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}$ Find a vector $\overrightarrow{\mathrm{d}}$ which is perpendicular to both $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$, and is such that $\overrightarrow{\mathrm{d}} \cdot \overrightarrow{\mathrm{c}}=21$. Solut...
Read More →Show that the vectors
Question: Show that the vectors $\frac{1}{7}(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+6 \hat{\mathrm{k}}), \frac{1}{7}(3 \hat{\mathrm{i}}-6 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})$ and $\frac{1}{7}(6 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}})$ are mutually perpendicular unit vectors. Solution:...
Read More →Solve this following
Question: If $\vec{a}=(\hat{\mathrm{i}}-\hat{\mathrm{j}}+7 \hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{b}}=(5 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\lambda \hat{\mathrm{k}})$ then find the value of $\lambda$ so that $(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}})$ and $(\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}})$ are orthogonal vectors. Solution:...
Read More →Solve this following
Question: i. If $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}$, show that $(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}})$ is perpendicular to $(\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}})$. ii. If $\overrightarrow{\mathrm{a}}=(5 \hat{\mathrm{i}}-\hat{\mathrm{j}}-3 \hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{b}}=(\hat{\mathrm{i}}+3 \hat{\mathr...
Read More →Solve this following
Question: Find the value of $\lambda$ for which $\vec{a}$ and $\vec{b}$ are perpendicular, where i. $\overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+\lambda \hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=(\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$ ii. $\overrightarrow{\mathrm{a}}=3 \hat{\mathrm{i}}-\hat{\mathrm{j}}+4 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=-\lambda \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ iii. $\overrightarrow{\mathrm{a...
Read More →Solve this following
Question: Find $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}$ when i. $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}-2 \hat{\mathrm{k}}$ ii. $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=-2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}$ iii. $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+5 \hat{\m...
Read More →Write a unit vector in the direction of
Question: Write a unit vector in the direction of $\overrightarrow{\mathrm{PQ}}$, where $\mathrm{P}$ and $\mathrm{Q}$ are the points $(1,3,0)$ and $(4,5,6)$ respectively. Solution:...
Read More →Solve this following
Question: If $\overrightarrow{\mathrm{AB}}=(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-3 \hat{\mathrm{k}})$ and $\mathrm{A}(1,2,-1)$ is the given point, find the coordinates of $\mathrm{B}$. Solution:...
Read More →