Mark against the correct answer in each of the following:
Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: The direction cosines of the vector $\overrightarrow{\mathrm{a}}=(-2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-5 \hat{\mathrm{k}})$are A. $-2,1,-5$ B. $\frac{1}{3}, \frac{-1}{6}, \frac{-5}{6}$ C. $\frac{2}{\sqrt{30}}, \frac{1}{\sqrt{30}}, \frac{5}{\sqrt{30}}$ D. $\frac{-2}{\sqrt{30}}, \frac{1}{\sqrt{30}}, \frac{-5}{\sqrt{30}}$ Solution:...
Read More →Mark against the correct answer in each of the following:
Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: A unit vector in the direction of the vector $\vec{a}=(2 \hat{i}-3 \hat{j}+6 \hat{k})_{\text {is }}$ is A. $\left(\hat{i}-\frac{3}{2} \hat{j}+3 \hat{k}\right)$ B. $\left(\frac{2}{5} \hat{i}-\frac{3}{5} \hat{j}+\frac{6}{5} \hat{k}\right)$ C. $\left(\frac{2}{7} \hat{i}-\frac{3}{7} \hat{j}+\frac{6}{7} \hat{k}\right)$ D. none of these Solution:...
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Question: If $\hat{\mathrm{a}}$ and $\hat{\mathrm{b}}$ are unit vectors such that $(\hat{\mathrm{a}}+\mathrm{b})$ is a unit vector, what is the angle between $\hat{\mathrm{a}}$ and $\hat{\mathrm{b}}$ ? Solution:...
Read More →Find the direction cosines of the vector
Question: Find the direction cosines of the vector $\overrightarrow{\mathrm{a}}=(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$ Solution:...
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Question: If $P(1,5,4)$ and $Q(4,1,-2)$ be the position vectors of two points $P$ and $Q$, find the direction ratios of $\overrightarrow{\mathrm{PQ}}$. Solution:...
Read More →Find the direction cosines of a vector which is equally inclined to the
Question: Find the direction cosines of a vector which is equally inclined to the $x-$ axis, $y-$ axis and $z$ - axis. Solution:...
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Question: When does$|\vec{a}+\vec{b}|=|\vec{a}|+|\vec{b}|$ hold? Solution:...
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Question: If $\theta$ is the angle between $\vec{a}$ and $\vec{b}$, and$|\vec{a} \cdot \vec{b}|=|\vec{a} \times \vec{b}|$ then what is the value of $\theta$ ? Solution:...
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Question: If $\vec{a}=(2 \hat{i}+6 \hat{j}+27 \hat{k})$ and $\overrightarrow{\mathrm{b}}=(\hat{\mathrm{i}}+\lambda \hat{\mathrm{j}}+\mu \hat{\mathrm{k}})$ are such that $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=\overrightarrow{0}$ then find the values of $\lambda$ and $\mu$. Solution:...
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Question: If $\overrightarrow{\mathrm{a}}=(-2 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}), \overrightarrow{\mathrm{b}}=(-2 \hat{\mathrm{j}}+4 \hat{\mathrm{j}}-2 \hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{c}}=(4 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}})$ then prove that $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}$ and $\vec{C}$ are coplanar. Solution:...
Read More →Find the volume of the parallelepiped whose edges are represented by the vectors
Question: Find the volume of the parallelepiped whose edges are represented by the vectors $\vec{a}=(2 \hat{i}-3 \hat{j}+4 \hat{k}), \vec{b}=(\hat{i}+2 \hat{j}-\hat{k})$ and $\overrightarrow{\mathrm{c}}=(3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})$ Solution:...
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Question: Write the value of $\hat{\mathrm{i}} \cdot(\hat{\mathrm{j}} \times \hat{\mathrm{k}})+\hat{\mathrm{j}} \cdot(\hat{\mathrm{i}} \times \hat{\mathrm{k}})+\hat{\mathrm{k}} \cdot(\hat{\mathrm{i}} \times \hat{\mathrm{j}})$ Solution:...
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Question: Find the value of $\lambda$ when the vectors $\vec{a}=(\hat{i}+\lambda \hat{j}+3 \hat{k})$ and $\overrightarrow{\mathrm{b}}=(3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+9 \hat{\mathrm{k}})$ are parallel. Solution:...
Read More →What conclusion can you draw about vectors
Question: What conclusion can you draw about vectors $\vec{a}$ and $\vec{b}$ when $\vec{a} \times \vec{b}=\overrightarrow{0}$ and $\vec{a} \cdot \vec{b}=0$ ? Solution:...
Read More →Find the angle between two vectors
Question: Find the angle between two vectors $\vec{a}$ and $\vec{b}$ with magnitudes 1 and 2 respectively, when $|\vec{a} \times \vec{b}|=\sqrt{3}$. Solution:...
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Question: If $\vec{a}=(\hat{i}-7 \hat{j}+7 \hat{k})$ and $\overrightarrow{\mathrm{b}}=(3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})$ then find $|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|$ Solution:...
Read More →Write the angle between two vectors
Question: Write the angle between two vectors $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ with magnitudes $\sqrt{3}$ and 2 respectively having $\vec{a} \cdot \vec{b}=\sqrt{6}$ Solution:...
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Question: Write the projection of the vector $(\hat{i}-\hat{j})$ on the vector $(\hat{i}+\hat{j})$ Solution:...
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Question: If $\overrightarrow{\mathrm{a}}=(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}), \overrightarrow{\mathrm{b}}=(4 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{c}}=(\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}})$ find a vector of magnitude 6 units which is parallel to the vector $(2 \overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}+3 \overrightarrow{\mathrm{c}})$ Solution:...
Read More →Write the vector of magnitude 15 units in the direction of vector
Question: Write the vector of magnitude 15 units in the direction of vector$(\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})$ Solution:...
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Question: If $\vec{a}=(2 \hat{i}+2 \hat{j}+3 \hat{k}), \vec{b}=(-\hat{i}+2 \hat{j}+\hat{k})$ and $\overrightarrow{\mathrm{c}}=(3 \hat{\mathrm{i}}+\hat{\mathrm{j}})$ are such that $(\vec{a}+\lambda \vec{b})$ is perpendicular to $\vec{C}$ then find the value of $\lambda$. Solution:...
Read More →Find a vector in the direction of
Question: Find a vector in the direction of$(2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+6 \hat{\mathrm{k}})$ which has magnitude 21 units. Solution:...
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Question: Find $\overrightarrow{\mathrm{a}} \cdot(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})$ when $\vec{a}=(2 \hat{i}+\hat{j}+3 \hat{k}), \vec{b}=(-\hat{i}+2 \hat{j}+\hat{k})$ and $\vec{c}=(3 \hat{i}+\hat{j}+2 \hat{k})$ Solution:...
Read More →Write the projection of the vector
Question: Write the projection of the vector $(7 \hat{\mathrm{i}}+\hat{\mathrm{j}}-4 \hat{\mathrm{k}})$ on the vector $(2 \hat{i}+6 \hat{j}+3 \hat{k})$ Solution:...
Read More →Write the projection of the vector
Question: Write the projection of the vector $(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})$ along the vector $\hat{\mathrm{j}}$ Solution:...
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