Using vector method, show that the points
Question: Using vector method, show that the points $A(4,5,1), B(0,-1,-1), C(3,9,4)$ and $D(-4,4,4)$ are coplanar. Solution:...
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Question: Find the value of $\lambda$ for which the four points with position vectors $(-\hat{\mathrm{j}}+\hat{\mathrm{K}})$ $(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}-\hat{\mathrm{k}}),(\hat{\mathrm{i}}+\lambda \hat{\mathrm{j}}+\hat{\mathrm{k}})$ and $(3 \hat{j}+3 \hat{k})$ are coplanar. Solution:...
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Question: Find the value of $\lambda$ for which the four points with position vectors$(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$ $(3 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}),(-2 \hat{\mathrm{i}}+\lambda \hat{\mathrm{j}}+\hat{\mathrm{k}})$ and $(6 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})$ are coplanar. Solution:...
Read More →Show that the four points with position vectors
Question: Show that the four points with position vectors $(6 \hat{\mathrm{i}}-7 \hat{\mathrm{j}}),(16 \hat{\mathrm{i}}-19 \hat{\mathrm{j}}-4 \hat{\mathrm{k}}),(3 \hat{\mathrm{j}}-6 \hat{\mathrm{k}})$ and $(2 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}+10 \hat{\mathrm{k}})$ are coplanar. Solution:...
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Question: Show that the four points with position vectors $(4 \hat{i}+8 \hat{j}+12 \hat{k}),(2 \hat{i}+4 \hat{j}+6 \hat{k})$ $(3 \hat{j}+5 \hat{j}+4 \hat{k})$ and $(5 \hat{\mathrm{i}}+8 \hat{\mathrm{j}}+5 \hat{\mathrm{k}})$ are coplanar. Solution:...
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Question: If the vectors $(a \hat{i}+a \hat{j}+c \hat{k}),(\hat{i}+\hat{k})$ and $(c \hat{i}+c \hat{j}+b \hat{k})$ be coplanar, show that $c^{2}=a b$ Solution: Then,...
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Question: Show that the vectors $\overrightarrow{\mathrm{a}}=(\hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\hat{\mathrm{k}}), \overrightarrow{\mathrm{b}}=(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}-\hat{\mathrm{k}})$ and $\vec{c}=(7 \hat{j}+3 \hat{k})$ are parallel to the same plane. \{HINT: Show that $[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}]=0_{\text {\} }}$ Solution:...
Read More →The volume of the parallelepiped whose edges are
Question: The volume of the parallelepiped whose edges are $(-12 \hat{\mathrm{i}}+\lambda \hat{\mathrm{k}}),(3 \hat{\mathrm{j}}-\hat{\mathrm{k}})$ and $(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-15 \hat{\mathrm{k}})$ is 546 cubic units. Find the value of $\lambda$. Solution:...
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Question: Find the value of $\lambda$ for which the vectors $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are coplanar, when i. $\overrightarrow{\mathrm{a}}=(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}), \overrightarrow{\mathrm{b}}=(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{c}}=(3 \hat{\mathrm{i}}+\lambda \hat{\mathrm{j}}+5 \hat{\mathrm{k}})$ ii. $\overrightarrow{\mathrm{a}}=\lambda \hat{\mathrm{i}}-10...
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Question: Show that the vectors $\vec{a}, \vec{b}, \vec{c}$ are coplanar, when i. $\vec{a}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=-2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-4 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}=\hat{\mathrm{i}}-3 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}$ ii. $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\ov...
Read More →Find the volume of the parallelepiped whose conterminous edges are represented by the vectors
Question: Find the volume of the parallelepiped whose conterminous edges are represented by the vectors+ i. $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{c}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}$ ii. $\overrightarrow{\mathrm{a}}=-3 \hat{\mathrm{i}}+7 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=-5 \hat{\mathrm{i}}+7 \hat{\m...
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Question: Find $\left[\begin{array}{lll}\vec{a} \vec{b} \vec{c}\end{array}\right]$, when i. $\vec{a}=2 \hat{i}+\hat{j}+3 \hat{k}, \vec{b}=-\hat{i}+2 \hat{j}+\hat{k}$ and $\vec{c}=3 \hat{i}+\hat{j}+2 \hat{k}$. ii. $\overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}=3 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}$ iii. $\overrightarrow{\mathrm...
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Question: i. $\left[\begin{array}{lll}\hat{\imath} \hat{\jmath} \hat{k}\end{array}\right]=\left[\begin{array}{lll}\hat{\jmath} \hat{k} \hat{\imath}\end{array}\right]=\left[\begin{array}{lll}\hat{k} \hat{\imath} \hat{\jmath}\end{array}\right]=1$ ii. $\left[\begin{array}{lll}\hat{\imath} \hat{k} \hat{\jmath}\end{array}\right]=\left[\begin{array}{lll}\hat{k} \hat{\jmath} \hat{\imath}\end{array}\right]=\left[\begin{array}{lll}\hat{\jmath} \hat{\imath} \hat{k}\end{array}\right]=-1$ Solution:...
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Question: If $\overrightarrow{\mathrm{a}}=(3 \hat{\mathrm{i}}-\hat{\mathrm{j}})$ and $\overrightarrow{\mathrm{b}}=(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-3 \hat{\mathrm{k}})$ then express $\overrightarrow{\mathrm{b}}$ in the form $\overrightarrow{\mathrm{b}}=\left(\overrightarrow{\mathrm{b}}_{1}+\overrightarrow{\mathrm{b}}_{2}\right)$, where $\overrightarrow{\mathrm{b}}_{1} \| \overrightarrow{\mathrm{a}}^{\text {and }} \overrightarrow{\mathrm{b}}_{2} \perp \overrightarrow{\mathrm{a}}^{2}$ Solution:...
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Question: If $\vec{a}$ and $\vec{b}$ are two vectors such that $|\vec{a}+\vec{b}|=|\vec{a}|$ then prove that vector $(2 \vec{a}+\vec{b})$ is perpendicular to the vector $\vec{b}$. Solution:...
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Question: If $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ are two unit vectors such that $|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|=\sqrt{3}$, find $(2 \overrightarrow{\mathrm{a}}-5 \overrightarrow{\mathrm{b}}) \cdot(3 \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}})$. Solution:...
Read More →If the position vectors of the verticesA
Question: If the position vectors of the verticesA, $B$ and $C$ of a $\triangle A B C$ be $(1,2,3),(-1,0,0)$ and $(0,1,2)$ respectively then find $\angle A B C$. Solution:...
Read More →Three vertices of a triangle are
Question: Three vertices of a triangle are $A(0,-1,-2), B(3,1,4)$ and $C(5,7,1)$. Show that it is a right - angled triangle. Also, find its other two angles. Solution:...
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Question: Show that the vectors $\vec{a}=(3 \hat{i}-2 \hat{j}+\hat{k}), \vec{b}=(\hat{i}-3 \hat{j}+5 \hat{k})$ and $\vec{c}=(2 \hat{i}+\hat{j}-4 \hat{k})$ form a right - angled triangle. Solution: Hence, the triangle is a right angled triangle at c...
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Question: Find the value of $\lambda$ for which the vectors $(2 \hat{i}+\lambda \hat{j}+3 \hat{k})$ and $(3 \hat{i}+2 \hat{j}-4 \hat{k})$ are perpendicular to each other. Solution:...
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Question: If $A(2,3,4), B(5,4,-1), C(3,6,2)$ and $D(1,2,0)$ be four points, show that $\overrightarrow{A B}$ is perpendicular to $\overrightarrow{C D}$. Solution:...
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Question: If $\overrightarrow{\mathrm{AB}}=(3 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})$ and the coordinates of $A$ are $(0,-2,-1)$, find the coordinates of $\mathrm{B}$. Solution:...
Read More →The dot products of a vector with the vector
Question: The dot products of a vector with the vector $(\hat{\mathrm{i}}+\hat{\mathrm{j}}-3 \hat{\mathrm{k}}),(\hat{\mathrm{i}}+3 \hat{\mathrm{j}}-2 \hat{\mathrm{k}})$ and $(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+4 \hat{\mathrm{k}})$ are 0,5 and 8 respectively. Find the vector. Solution:...
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Question: If $\hat{\mathrm{a}}$ and $\hat{\mathrm{b}}$ are unit vectors inclined at an angle $\theta$ then prove that: i. $\cos \frac{\theta}{2}=\frac{1}{2}|\hat{a}+\hat{b}|$ ii. $\tan \frac{\theta}{2}=\frac{|\hat{\mathrm{a}}-\hat{\mathrm{b}}|}{|\hat{\mathrm{a}}+\hat{\mathrm{b}}|}$ Solution:...
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