Find the sum of the vectors
Question: Find the sum of the vectors$\overrightarrow{\mathrm{a}}=(\hat{\mathrm{i}}-2 \hat{\mathrm{j}}), \overrightarrow{\mathrm{b}}=(2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}})$ and $\overrightarrow{\mathrm{c}}=(2 \hat{\mathrm{i}}+3 \hat{\mathrm{k}})$ Solution:...
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Question: Find the sum of the vectors $\overrightarrow{\mathrm{a}}=(\hat{\mathrm{i}}-3 \hat{\mathrm{k}}), \overrightarrow{\mathrm{b}}=(2 \hat{\mathrm{j}}-\hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{c}}=(2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})$ Solution:...
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Question: If $\vec{a}$ is a unit vector such that $(\vec{x}-\vec{a}) \cdot(\vec{x}+\vec{a})=15$ find $|\vec{x}|$. Solution: $(\vec{x}-\vec{a})(\vec{x}+\vec{a})=15$ $\left.\Rightarrow\left|\vec{x}_{\left.\right|^{2}}-\right| \vec{a}\right|^{2}=15$ $\Rightarrow|\vec{x}|^{2}=|\vec{a}|^{2}+15$ Now, a is a unit vector, $\Rightarrow|\vec{a}|=1$ $\Rightarrow|\vec{x}|^{2}=1^{2}+15$ $\Rightarrow|\vec{x}|^{2}=16$ $\Rightarrow|\vec{x}|=4$ Ans: $\left.\right|^{2} \mid=4$...
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Question: If $\vec{a}$ and $\vec{b}$ are perpendicular vectors such that $|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|=13$ and $|\vec{a}|=5$ find the value of $|\overrightarrow{\mathrm{b}}|$ Solution:...
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Question: Find the value of $\lambda$ when the projection of $\vec{a}=(\lambda \hat{i}+\hat{j}+4 \hat{k})$ on $\overrightarrow{\mathrm{b}}=(2 \hat{\mathrm{i}}+6 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$ is 4 units. Solution:...
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Question: Find the value of $p$ for which the vectors $\vec{a}=(3 \hat{i}+2 \hat{j}+9 \hat{k})$ and $\overrightarrow{\mathrm{b}}=(\hat{\mathrm{i}}-2 p \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$ are parallel. Solution:...
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Question: Write the value of $\lambda$ so that the vectors $\vec{a}=(2 \hat{i}+\lambda \hat{j}+\hat{k})$ and $\overrightarrow{\mathrm{b}}=(\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$ are perpendicular to each other. Solution:...
Read More →Write a unit vector in the direction of the sum of the vectors
Question: Write a unit vector in the direction of the sum of the vectors $\vec{a}=(2 \hat{i}+2 \hat{j}-5 \hat{k})$ and $\overrightarrow{\mathrm{b}}=(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-7 \hat{\mathrm{k}})$ Solution:...
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Question: If $\overrightarrow{\mathrm{a}}=\mathrm{x} \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\mathrm{z} \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}-\mathrm{y} \hat{\mathrm{j}}+\hat{\mathrm{k}}$ are two equal vectors the $x+y+z=?$ Solution:...
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Question: Find the value of $\lambda$ for which the points $A(3,2,1), B(4, \lambda, 5), C(4,2,-2)$ and $D(6,5,-1)$ are coplanar. Solution:...
Read More →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 $(3 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$ are coplanar. Solution:...
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Question: Find the value of $\lambda$ for which the four points with position vectors $(\hat{i}+2 \hat{j}+3 \hat{k})$ $(3 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}),(-2 \hat{\mathrm{i}}+\lambda \hat{\mathrm{j}}+\hat{\mathrm{k}})$ 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$(6 \hat{\mathrm{i}}-7 \hat{\mathrm{j}}),(16 \hat{\mathrm{i}}-19 \hat{\mathrm{j}}-4 \hat{\mathrm{k}}),(3 \hat{\mathrm{j}}-6 \hat{\mathrm{k}})$ $(2 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}+10 \hat{\mathrm{k}})$ are coplanar. Solution:...
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Question: Show that the four points with position vectors $(4 \hat{\mathrm{i}}+8 \hat{\mathrm{j}}+12 \hat{\mathrm{k}}),(2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+6 \hat{\mathrm{k}})$ $(3 \hat{j}+5 \hat{j}+4 \hat{k})$ 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 $(c \hat{\mathrm{i}}+\mathrm{c} \hat{\mathrm{j}}+\mathrm{b} \hat{\mathrm{k}})$ be coplanar, show that $c^{2}=a b .$ Solution:...
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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 $\overrightarrow{\mathrm{c}}=(7 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$ are parallel to the same plane. \{HINT: Show that $\left[\begin{array}{lll}\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\end{array}\right]=0_{\}}$ Solution:...
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Question: If $\vec{a}$ and $\vec{b}$ are perpendicular vectors such that $|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|=13$ and $|\vec{a}|=5$ find the value of $|\vec{b}| .$ Solution:...
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Question: Find the value of $\lambda$ when the projection of $\vec{a}=(\lambda \hat{i}+\hat{j}+4 \hat{k})$ on $\overrightarrow{\mathrm{b}}=(2 \hat{\mathrm{i}}+6 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$ is 4 units. Solution:...
Read More →Find the value of p for which the vectors
Question: Find the value of $p$ for which the vectors $\vec{a}=(3 \hat{i}+2 \hat{j}+9 \hat{k})$ and $\overrightarrow{\mathrm{b}}=(\hat{\mathrm{i}}-2 p \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$ are parallel. Solution:...
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Question: Write the value of $\lambda$ so that the vectors $\vec{a}=(2 \hat{i}+\lambda \hat{j}+\hat{k})$ and $\overrightarrow{\mathrm{b}}=(\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$ are perpendicular to each other. Solution:...
Read More →Write a unit vector in the direction of the sum of the vectors
Question: Write a unit vector in the direction of the sum of the vectors $\overrightarrow{\mathrm{a}}=(2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-5 \hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{b}}=(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-7 \hat{\mathrm{k}})$ Solution:...
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Question: If $\vec{a}=x \hat{i}+2 \hat{j}-z \hat{k}$ and $\overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}-\mathrm{y} \hat{\mathrm{j}}+\hat{\mathrm{k}}$ are two equal vectors the $x+y+z=?$ Solution:...
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Question: Find the value of $\lambda$ for which the points $A(3,2,1), B(4, \lambda, 5), C(4,2,-2)$ and $D(6,5,-1)$ are coplanar. Solution: $=-9-14+7 \lambda-12$...
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