Find the position vector of the mid-point
Question: Find the position vector of the mid-point of the vector joining the points $A(3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+6 \hat{\mathrm{k}})$ and $B(\hat{i}+4 \hat{j}-2 \hat{k})$. Solution:...
Read More →Find the position vector of a point
Question: Find the position vector of a point $\mathrm{R}$ which divides the line joining $A(-2,1,3)$ and $B(3,5,-2)$ in the ratio $2: 1$ (i) internally (ii) externally. Solution:...
Read More →The position vectors of two points
Question: The position vectors of two points $A$ and $B$ are $(2 \vec{a}+\vec{b})$ and $(\vec{a}-3 \vec{b})$ respectively. Find the position vector of a point $C$ which divides $A B$ externally in the ratio $1: 2$. Also, show that $A$ is the mid-point of the line segment CB. Solution:...
Read More →Find the position vector of the point which divides the join of the points
Question: Find the position vector of the point which divides the join of the points $(2 \vec{a}-3 \vec{b})$ and $(3 \vec{a}-2 \vec{b})$ (i) internally and (ii) externally in the ratio $2: 3$. Solution:...
Read More →Using vector method, show that the points
Question: Using vector method, show that the points $A(1,-1,0), B(4,-3,1)$ and $C(2,-4,5)$ are the vertices of a rightangled triangle. Solution:...
Read More →Solve this following
Question: Show that the points $A, B$ and $C$ having position vectors $(3 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}-4 \hat{\mathrm{k}}),(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})$ and $(\hat{\mathrm{i}}-3 \hat{\mathrm{j}}-5 \hat{\mathrm{k}})$ respectively, form the vertices of a right-angled triangle. Solution:...
Read More →If the position vectors of the vertices
Question: If the position vectors of the vertices $A, B$ and $C$ of a $\triangle A B C$ be $(\hat{i}+2 \hat{j}+3 \hat{k}),(2 \hat{i}+3 \hat{j}+\hat{k})$ and $(3 \hat{i}+\hat{j}+2 \hat{k})$ respectively, prove that $\triangle A B C$ is equilateral. Solution:...
Read More →The position vectors of the points
Question: The position vectors of the points $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are $(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}),(3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}})$ and $(\hat{\mathrm{i}}+4 \hat{\mathrm{j}}-3 \hat{\mathrm{k}})$ respectively. Show that the points A, B and C are collinear. Solution:...
Read More →Solve this following
Question: Show that the points $A, B$ and $C$ having position vectors $(\hat{i}+2 \hat{j}+7 \hat{k}),(2 \hat{i}+6 \hat{j}+3 \hat{k})$ and $(3 \hat{i}+10 \hat{j}-3 \hat{k})$ respectively, are collinear. Solution:...
Read More →Find the direction ratios and the direction cosines of the vector joining the points
Question: Find the direction ratios and the direction cosines of the vector joining the points $A(2,1,-2)$ and $B(3,5,-4)$. Solution:...
Read More →Find the direction ratios and direction cosines of the vector
Question: Find the direction ratios and direction cosines of the vector $\vec{a}=(5 \hat{i}-3 \hat{j}+4 \hat{k})$. Solution:...
Read More →Solve this following
Question: If $A(-2,1,2)$ and $B(2,-1,6)$ are two given points, find a unit vector in the direction of $\overrightarrow{A B}$. Solution:...
Read More →Solve this following
Question: If $\overrightarrow{\mathrm{a}}=(\hat{\mathrm{i}}-2 \hat{\mathrm{j}}), \overrightarrow{\mathrm{b}}=(2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}})$ and $\overrightarrow{\mathrm{c}}=(2 \hat{\mathrm{i}}+3 \hat{\mathrm{k}})$, find $(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}})$ Solution:...
Read More →Find a vector of magnitude 21 units in the direction of the vector
Question: Find a vector of magnitude 21 units in the direction of the vector $(2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+6 \hat{\mathrm{k}})$. Solution:...
Read More →Find a vector of magnitude 8 units in the direction of the vector
Question: Find a vector of magnitude 8 units in the direction of the vector $(5 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})$. Solution:...
Read More →Find a vector of magnitude 9 units in the direction of the vector
Question: Find a vector of magnitude 9 units in the direction of the vector $(-2 \hat{i}+\hat{j}+2 \hat{k})$. Solution:...
Read More →Solve this following
Question: If $\overrightarrow{\mathrm{a}}=(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{b}}=(2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+9 \hat{\mathrm{k}})$ then find a unit vector parallel to $(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}})$. Solution:...
Read More →Solve this following
Question: If $\overrightarrow{\mathrm{a}}=(3 \hat{\mathrm{i}}+\hat{\mathrm{j}}-5 \hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{b}}=(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}})$ then find a unit vector in the direction of $(\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}})$. Solution:...
Read More →Solve this following
Question: If $\overrightarrow{\mathrm{a}}=(-\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{b}}=(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})$ then find the unit vector in the direction of $(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}})$. Solution:...
Read More →Solve this following
Question: If $\overrightarrow{\mathrm{a}}=(2 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}+5 \hat{\mathrm{k}})$ then find the value of $\lambda$ so that $\lambda \overrightarrow{\mathrm{a}}$ may be a unit vector. Solution:...
Read More →Find a unit vector in the direction of the vector:
Question: Find a unit vector in the direction of the vector: A. $(3 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-5 \hat{\mathrm{k}})$ B. $(3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+6 \hat{\mathrm{k}})$ C. $(\hat{\mathrm{i}}+\hat{\mathrm{k}})$ D. $(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+2 \hat{\mathrm{k}})$ Solution:...
Read More →Write down the magnitude of each of the following vectors:
Question: Write down the magnitude of each of the following vectors: A. $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}$ B. $\overrightarrow{\mathrm{b}}=5 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}$ C. $\overrightarrow{\mathrm{c}}=\left(\frac{1}{\sqrt{3}} \hat{\mathrm{i}}-\frac{1}{\sqrt{3}} \hat{\mathrm{j}}+\frac{1}{\sqrt{3}} \hat{\mathrm{k}}\right)$ D. $\overrightarrow{\mathrm{d}}=(\sqrt{2} \hat{\mathrm{i}}+\sqrt{3} \hat{\mathrm{j}}-\sqrt{5} \hat{...
Read More →Mark against the correct answer in the following:
Question: Mark $(\sqrt{)}$ against the correct answer in the following: The general solution of the $D E \frac{d y}{d x}+\frac{y}{x}=x^{2}$ is A. $x y=x^{4}+\mathrm{C}$ B. $4 x y=x^{4}+C$ C. $3 x y=x^{3}+\mathrm{C}$ D. None of these Solution:...
Read More →Mark against the correct answer in the following:
Question: Mark $(\sqrt{ })$ against the correct answer in the following: The general solution of the $\mathrm{DE} \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y} \cot \mathrm{x}=2 \cos \mathrm{x}$ is A. $(y+\sin x) \sin x=C$ B. $(y+\cos x) \sin x=\mathrm{C}$ C. $(y-\sin x) \sin x=\mathrm{C}$ D. None of these Solution:...
Read More →Mark against the correct answer in the following:
Question: Mark $(\sqrt{ })$ against the correct answer in the following: The general solution of the $D E \frac{d y}{d x}+y \tan x=\sec x$ is A. $y=\sin x-C \cos x$ B. $y=\sin x+C \cos x$ C. $y=\cos x-C \sin x$ D. None of these Solution:...
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