Question 11 Mark
Write the position vector of the point which divides the join of points with position vectors $3\overrightarrow{\text{a}} - 2\overrightarrow{\text{b}} \text{and } 2\overrightarrow{\text{a}} + 3\overrightarrow{\text{b}}$ in the ratio $2:1.$
Answer$\frac{2(2\overrightarrow{\text{a}}+3\overrightarrow{\text{b}})+1 (3\overrightarrow{\text{a}} - 2\overrightarrow{\text{b}})}{2 +1} $
$= \frac{7}{3}\overrightarrow{\text{a}} + \frac{4}{3}\overrightarrow{\text{b}}$
View full question & answer→Question 21 Mark
Write the number of vectors of unit length perpendicular to both the vectors $\overrightarrow{\text{a}} = 2\hat{\text{i}} + \hat{j} + 2\hat{\text{k}} \text{ and} \overrightarrow{\text{b}} = \hat{\text{j}} + \hat{\text{k}}.$
View full question & answer→Question 31 Mark
Find a vector of magnitude $\sqrt{171}$ which is perpendicular to both of the vectors $\overrightarrow{\text{a}} = \hat{\text{i}} + 2\hat{\text{j}} - 3\hat{\text{k}}$ $\text{and} \overrightarrow{\text{b}} = 3\hat{\text{i}} - \hat{\text{j}} + 2\hat{\text{k}}.$
AnswerVector Perpendicular to $\overrightarrow{\text{a}} \text{and} \overrightarrow{\text{b}} = \frac{\overrightarrow{\text{a}}\times\overrightarrow{\text{b}}}{\overrightarrow|{\text{a}}\times\overrightarrow{\text{b}}|}$
[Finding using]
Required Vector $ = \hat{\text{i}} - 11\hat{\text{j}} - 7\hat{\text{k}}$
View full question & answer→Question 41 Mark
If $\overrightarrow{\text{a}}$and$\overrightarrow{\text{b}}$ are perpendicular vectors,|$\overrightarrow{\text{a}}$+$\overrightarrow{\text{b}}$|= 13 and |$\overrightarrow{\text{a}}$| = 5 find the value of|$\overrightarrow{\text{b}}$|.
View full question & answer→Question 51 Mark
Write the position vector of the point which divides the join of points with position vectors $3\overrightarrow{\text{a}} - 2\overrightarrow{\text{b}} \text{and } 2\overrightarrow{\text{a}} + 3\overrightarrow{\text{b}}$ in the ratio $2:1.$
Answer$\frac{2(2\overrightarrow{\text{a}}+3\overrightarrow{\text{b}})+1 (3\overrightarrow{\text{a}} - 2\overrightarrow{\text{b}})}{2 +1} $
$= \frac{7}{3}\overrightarrow{\text{a}} + \frac{4}{3}\overrightarrow{\text{b}}$
View full question & answer→Question 61 Mark
Write the number of vectors of unit length perpendicular to both the vectors $\overrightarrow{\text{a}} = 2\hat{\text{i}} + \hat{j} + 2\hat{\text{k}} \text{ and} \overrightarrow{\text{b}} = \hat{\text{j}} + \hat{\text{k}}.$
View full question & answer→Question 71 Mark
Find a vector of magnitude $\sqrt{171}$ which is perpendicular to both of the vectors $\overrightarrow{\text{a}} = \hat{\text{i}} + 2\hat{\text{j}} - 3\hat{\text{k}}$ $\text{and} \overrightarrow{\text{b}} = 3\hat{\text{i}} - \hat{\text{j}} + 2\hat{\text{k}}.$
AnswerVector Perpendicular to $\overrightarrow{\text{a}} \text{and} \overrightarrow{\text{b}} = \frac{\overrightarrow{\text{a}}\times\overrightarrow{\text{b}}}{\overrightarrow|{\text{a}}\times\overrightarrow{\text{b}}|}$
[Finding using]
Required Vector $ = \hat{\text{i}} - 11\hat{\text{j}} - 7\hat{\text{k}}$
View full question & answer→Question 81 Mark
Find a vector$\overrightarrow{\text{a}}$ magnitude,$\frac{5}{\sqrt{2}}$ making an angle of with$\frac{\pi}{4}$ x-axis, $\frac{\pi}{2}$with y-axis and an acute angle $\theta$ with z-axis.
Answer$\overrightarrow{\text{a}}=5\hat{\text{i}}+5\hat{\text{k}}$
View full question & answer→Question 91 Mark
Write the position vector of the point which divides the join of points with position vectors $3\overrightarrow{\text{a}} - 2\overrightarrow{\text{b}} \text{and } 2\overrightarrow{\text{a}} + 3\overrightarrow{\text{b}}$ in the ratio $2:1.$
Answer$\frac{2(2\overrightarrow{\text{a}}+3\overrightarrow{\text{b}})+1 (3\overrightarrow{\text{a}} - 2\overrightarrow{\text{b}})}{2 +1} $
$= \frac{7}{3}\overrightarrow{\text{a}} + \frac{4}{3}\overrightarrow{\text{b}}$
View full question & answer→Question 101 Mark
Write the number of vectors of unit length perpendicular to both the vectors $\overrightarrow{\text{a}} = 2\hat{\text{i}} + \hat{j} + 2\hat{\text{k}} \text{ and} \overrightarrow{\text{b}} = \hat{\text{j}} + \hat{\text{k}}.$
View full question & answer→Question 111 Mark
Find a vector of magnitude $\sqrt{171}$ which is perpendicular to both of the vectors $\overrightarrow{\text{a}} = \hat{\text{i}} + 2\hat{\text{j}} - 3\hat{\text{k}}$ $\text{and} \overrightarrow{\text{b}} = 3\hat{\text{i}} - \hat{\text{j}} + 2\hat{\text{k}}.$
AnswerVector Perpendicular to $\overrightarrow{\text{a}} \text{and} \overrightarrow{\text{b}} = \frac{\overrightarrow{\text{a}}\times\overrightarrow{\text{b}}}{\overrightarrow|{\text{a}}\times\overrightarrow{\text{b}}|}$
[Finding using]
Required Vector $ = \hat{\text{i}} - 11\hat{\text{j}} - 7\hat{\text{k}}$
View full question & answer→Question 121 Mark
Find $|\overrightarrow{\text{x}}|,$ if for a unit vector $\overrightarrow{\text{a}},(\overrightarrow{\text{x}} - \overrightarrow{\text{a}}).(\overrightarrow{\text{x}} + \overrightarrow{\text{a}}) = 15.$
AnswerGiven $(\overrightarrow{\text{x}} - \overrightarrow{\text{a}}).(\overrightarrow{\text{x}} + \overrightarrow{\text{a}})= 15 $
$\Rightarrow(\overrightarrow{\text{x}})^{2} - (\overrightarrow{\text{a}})^{2} = 15 $
$\Rightarrow\overrightarrow{\text{x}}.\overrightarrow{\text{x}} - \overrightarrow{\text{a}}.\overrightarrow{\text{a}} = 15 \Rightarrow|\overrightarrow{\text{x}}|^{2} - |\overrightarrow{\text{a}}|^{2} = 15 $
$\Rightarrow|\overrightarrow{\text{x}}|^{2} - 1 = 15 \Rightarrow|\overrightarrow{\text{x}}|^{2} = 16$
$\Rightarrow|\overrightarrow{\text{x}}| = 4 [\because - \text{ ve value is not acceptable}].$
View full question & answer→Question 131 Mark
P and Q are two points with position vectors 3$\overrightarrow{\text{a}}$ - 2$\overrightarrow{\text{b}}$ and $\overrightarrow{\text{a}} + \overrightarrow{\text{b}}$respectively. Write the position vector of a point R which divides the line segment PQ in the ratio 2:1 externally.
AnswerIf $\overrightarrow{\text{r}}$is the position vector of R then by section formula
$\overrightarrow{\text{r}} = \frac{2(\overrightarrow{\text{a}} + \overrightarrow{\text{b}} )-1.(3\overrightarrow{\text{a}} - 2 \overrightarrow{\text{b}})}{2-1}$
$ = \frac{2\overrightarrow{\text{a}}+ 2\overrightarrow{\text{b}} - 3\overrightarrow{\text{a}} + 2 \overrightarrow{\text{b}}}{1} = 4 \overrightarrow{\text{b}} - \overrightarrow{\text{a}}.$ View full question & answer→Question 141 Mark
Write the value of $(\hat{\text{i}}\times\hat{\text{j}})\cdot\hat{\text{k}}+\hat{\text{i}}\cdot\hat{\text{j} }$.
View full question & answer→Question 151 Mark
Find the scalar components of the vector $\overrightarrow{\text{AB}}$ with initial point A(2, 1) and terminal point B(- 5, 7).
View full question & answer→Question 161 Mark
Write the projection of the vector $\hat{\text{i}}-\hat{\text{j}}$on the vector $\hat{\text{i}}+\hat{\text{j}}$.
View full question & answer→Question 171 Mark
Write a vector of magnitude 9 units in the direction of vector$-2\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}}$
Answer$-6\hat{\text{i}}+3\hat{\text{j}}+6\hat{\text{k}}$
View full question & answer→Question 181 Mark
Find the value of p if $(2\hat{\text{i}}+6\hat{\text{j}}+27\hat{\text{k)}}\times(\hat{\text{i}}+3\hat{\text{j}}+\text{p}\hat{\text{k}})=\vec{0}.$
View full question & answer→Question 191 Mark
If $\vec{\text{p}}$ is a unit vector and $(\vec{\text{x}}-\vec{\text{p}})\cdot(\vec{\text{x}}+\vec{\text{p}})=80,$ then find $|\vec{\text{x}}|.$
View full question & answer→Question 201 Mark
If $\vec{\text{a}}=\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}}\text{ and }\vec{\text{b}}=3\hat{\text{i}}+\hat{\text{j}}-5\hat{\text{k}}$ find a unit vector in the direction of $\vec{\text{a}}-\vec{\text{b}}$.
Answer$\vec{\text{a}}-\vec{\text{b}}=-2\hat{\text{i}}+\hat{\text{j}}+4\hat{\text{k}}$
Unit vector in the direction of $(\vec{\text{a}}-\vec{\text{b}})$ $=\frac{1}{\sqrt{21}}[-2\hat{\text{i}}+\hat{\text{j}}+4\hat{\text{k}}]$.
View full question & answer→Question 211 Mark
If $|\vec{\text{a}}|=\sqrt{3},|\vec{\text{b}}|=2\text{ and }\vec{\text{a}}\cdot\vec{\text{b}}=3$find the angle between $\vec{\text{a}}$and $\vec{\text{b}}$.
Answer$\theta=\cos^{-1}\Bigg(\frac{\sqrt{3}} {2}\Bigg)=\frac{\pi}{6}.$
View full question & answer→Question 221 Mark
If $\vec{\text{P}}$ (1, 5, 4) and $\vec{\text{Q}}$ (4, 1, -2), find the direction ratios of $\vec{\text{PQ}}$.
View full question & answer→Question 231 Mark
If $\vec{\text{a}}, \text{ }\vec{\text{b}},\text{ } \vec{\text{c}}$ are unit vectors such that $\vec{\text{a}}, \text{ }\vec{\text{b}}, \text{ }\vec{\text{c}}\text{ }=\vec{0},$ then write the value of $\vec{\text{a}} \text{ . }\vec{\text{b}} + \vec{\text{b}} \text{ . } \text{ }\vec{\text{c}} +\vec{\text{c}} . \vec{\text{a}}\text{ }.$
Answer$(\vec{\text{a}} + \vec{\text{b}} + \vec{\text{c}}) . (\vec{\text{a}} + \vec{\text{b}} + \vec{\text{c}}) = 0$
$\Rightarrow |\vec{\text{a}}|^{2} + |\vec{\text{b}}|^{2} + |\vec{\text{c}}|^{2} + 2 (\vec{\text{a}}.\vec{\text{b}} + \vec{\text{b}}.\vec{\text{c}} + \vec{\text{c}}.\vec{\text{a}}) = 0$
$\Rightarrow \vec{\text{a}}.\vec{\text{b}} + \vec{\text{b}}.\vec{\text{c}} + \vec{\text{c}}.\vec{\text{a}} = -\frac{3}{2}$
View full question & answer→Question 241 Mark
If $\big|\vec{\text{a}} \times \vec{\text{b}}\big|^{2} + \big|\vec{\text{a}} . \vec{\text{b}}\big|^{2} = 400 \text{ and } \big|\vec{\text{a}}\big| = 5,$ then write the value of $\big|\vec{\text{b}}\big|.$
Answer$\text{a}^{2} \text{b}^{2} \sin^{2} \theta + \text{a}^{2} \text{b}^{2} \cos^{2} \theta = 400$
$\Rightarrow |\vec{\text{b}}| = 4$
View full question & answer→Question 251 Mark
If $\vec{\text{a}}, \text{ }\vec{\text{b}},\text{ } \vec{\text{c}}$ are unit vectors such that $\vec{\text{a}}, \text{ }\vec{\text{b}}, \text{ }\vec{\text{c}}\text{ }=\vec{0},$ then write the value of $\vec{\text{a}} \text{ . }\vec{\text{b}} + \vec{\text{b}} \text{ . } \text{ }\vec{\text{c}} +\vec{\text{c}} . \vec{\text{a}}\text{ }.$
Answer$(\vec{\text{a}} + \vec{\text{b}} + \vec{\text{c}}) . (\vec{\text{a}} + \vec{\text{b}} + \vec{\text{c}}) = 0$
$\Rightarrow |\vec{\text{a}}|^{2} + |\vec{\text{b}}|^{2} + |\vec{\text{c}}|^{2} + 2 (\vec{\text{a}}.\vec{\text{b}} + \vec{\text{b}}.\vec{\text{c}} + \vec{\text{c}}.\vec{\text{a}}) = 0$
$\Rightarrow \vec{\text{a}}.\vec{\text{b}} + \vec{\text{b}}.\vec{\text{c}} + \vec{\text{c}}.\vec{\text{a}} = -\frac{3}{2}$
View full question & answer→Question 261 Mark
If $\big|\vec{\text{a}} \times \vec{\text{b}}\big|^{2} + \big|\vec{\text{a}} . \vec{\text{b}}\big|^{2} = 400 \text{ and } \big|\vec{\text{a}}\big| = 5,$ then write the value of $\big|\vec{\text{b}}\big|.$
Answer$\text{a}^{2} \text{b}^{2} \sin^{2} \theta + \text{a}^{2} \text{b}^{2} \cos^{2} \theta = 400$
$\Rightarrow |\vec{\text{b}}| = 4$
View full question & answer→Question 271 Mark
If $\big|\vec{\text{a}} \times \vec{\text{b}}\big|^{2} + \big|\vec{\text{a}} . \vec{\text{b}}\big|^{2} = 400 \text{ and } \big|\vec{\text{a}}\big| = 5,$ then write the value of $\big|\vec{\text{b}}\big|.$
Answer$\text{a}^{2} \text{b}^{2} \sin^{2} \theta + \text{a}^{2} \text{b}^{2} \cos^{2} \theta = 400$
$\Rightarrow |\vec{\text{b}}| = 4$
View full question & answer→Question 281 Mark
If $\vec{\text{a}}, \text{ }\vec{\text{b}},\text{ } \vec{\text{c}}$ are unit vectors such that $\vec{\text{a}}, \text{ }\vec{\text{b}}, \text{ }\vec{\text{c}}\text{ }=\vec{0},$ then write the value of $\vec{\text{a}} \text{ . }\vec{\text{b}} + \vec{\text{b}} \text{ . } \text{ }\vec{\text{c}} +\vec{\text{c}} . \vec{\text{a}}\text{ }.$
Answer$(\vec{\text{a}} + \vec{\text{b}} + \vec{\text{c}}) . (\vec{\text{a}} + \vec{\text{b}} + \vec{\text{c}}) = 0$
$\Rightarrow |\vec{\text{a}}|^{2} + |\vec{\text{b}}|^{2} + |\vec{\text{c}}|^{2} + 2 (\vec{\text{a}}.\vec{\text{b}} + \vec{\text{b}}.\vec{\text{c}} + \vec{\text{c}}.\vec{\text{a}}) = 0$
$\Rightarrow \vec{\text{a}}.\vec{\text{b}} + \vec{\text{b}}.\vec{\text{c}} + \vec{\text{c}}.\vec{\text{a}} = -\frac{3}{2}$
View full question & answer→Question 291 Mark
Find the vector equation of a plane which is at a distance of 5 units from the origin and its normal vector is $2\hat{\text{i}} - 3\hat{\text{j}} + 6\hat{\text{k}}.$
Answer$\overrightarrow{\text{r}}. \frac{(2\hat{\text{i}} - 3\hat{\text{j}} + 6\hat{\text{k})}}{7} = 5$
View full question & answer→Question 301 Mark
Find the position vector of a point which divides the join of points with position vectors $\overrightarrow{\text{a}} - \overrightarrow{\text{2b }} $ and $\overrightarrow{\text{2a}} +\overrightarrow{\text{b}}$externally in the ratio 2:1.
AnswerGetting position vector as $2(\overrightarrow{2\text{a}} +\overrightarrow{\text{b}}) - 1 (\overrightarrow{\text{a}} - 2\overrightarrow{\text{b}})$
$= \overrightarrow{3\text{a}} + \overrightarrow{4\text{b}}$
View full question & answer→Question 311 Mark
The two vectors $\hat{\text{j}} + \hat{\text{k}}$ and $3\hat{\text{i}} - \hat{\text{j}} + 4\hat{\text{k}}$ represent the two sides AB and AC, respectively of a $\triangle \text{ABC},$ Find the length of the median through A.
Answer$\overrightarrow{\text{AD}} = \overrightarrow{\text{AB}} + \frac{1}{2}[\overrightarrow{\text{AC}} - \overrightarrow{\text{AB}}] = \frac{1}{2} (\overrightarrow{\text{AC}} + \overrightarrow{\text{AB}})$
$|\overrightarrow{\text{AD}}| =\frac{1}{2} |3\hat{\text{i}} + 5\hat{\text{k}}| = \frac{1}{2}\sqrt{34}$
View full question & answer→Question 321 Mark
$\text{If} \overrightarrow{\text{a}} = 7\hat{\text{i}} + \hat{\text{j}} - 4\hat{\text{k}}$ and $\overrightarrow{\text{b}} = 2\hat{\text{i}} - 6\hat{\text{j}} + 3\hat{\text{k}},$ then find projection of $\overrightarrow{\text{a}}\text{on} \overrightarrow{\text{b}}.$
Answer$\text{p} = \frac{\overrightarrow{a}. \overrightarrow{b}}{\overrightarrow{|b}|} = \frac{8}{7}$
View full question & answer→Question 331 Mark
Write a unit vector in the direction of the sum of the vectors $\vec{\text{a}}=2\hat{\text{i}}+2\hat{\text{j}}-5\hat{\text{k}}\ \text{and}\ \vec{\text{b}}=2\hat{\text{i}}+\hat{\text{j}}-7\hat{\text{k}}$
AnswerThe given vectors are and $\vec{\text{b}}=2\hat{\text{i}}+\hat{\text{j}}-7\hat{\text{k}}$
Now, $\vec{\text{a}}+\vec{\text{b}}=4\hat{\text{i}}+3\hat{\text{j}}-12\hat{\text{k}}$
$\Rightarrow|\vec{\text{a}}+\vec{\text{b}}|=\sqrt{4^2+3^2+(-12)^2}$
$=\sqrt{16+9+144}=\sqrt{169}=13$
$\therefore$ The unit vector in the direction of $\vec{\text{a}}+\vec{\text{b}} \text{ is}\frac{\vec{\text{a}}+\vec{\text{b}}}{|\vec{\text{a}}+\vec{\text{a}}|}=\frac{1}{13}(4\hat{\text{i}}+3\hat{\text{j}}-12\hat{\text{k}})$
$\frac{4}{13}\hat{\text{i}}+\frac{3}{13}\hat{\text{j}}-\frac{12}{13}\hat{\text{k}}$
View full question & answer→Question 341 Mark
Find the projection of the vector $\hat{\text{i}} + 3\hat{\text{j}} + 7\hat{\text{k}}\text{ on the vector } 2\hat{\text{i}} -3\hat{\text{j}} + 6\hat{\text{k}}.$
View full question & answer→Question 351 Mark
The two vectors $\hat{\text{j}} + \hat{\text{k}}$ and $3\hat{\text{i}} - \hat{\text{j}} + 4\hat{\text{k}}$ represent the two sides AB and AC, respectively of a $\triangle \text{ABC},$ Find the length of the median through A.
Answer$\overrightarrow{\text{AD}} = \overrightarrow{\text{AB}} + \frac{1}{2}[\overrightarrow{\text{AC}} - \overrightarrow{\text{AB}}] = \frac{1}{2} (\overrightarrow{\text{AC}} + \overrightarrow{\text{AB}})$
$|\overrightarrow{\text{AD}}| =\frac{1}{2} |3\hat{\text{i}} + 5\hat{\text{k}}| = \frac{1}{2}\sqrt{34}$
View full question & answer→Question 361 Mark
Find the position vector of a point which divides the join of points with position vectors $\overrightarrow{\text{a}} - \overrightarrow{\text{2b }} $ and $\overrightarrow{\text{2a}} +\overrightarrow{\text{b}}$externally in the ratio 2:1.
AnswerGetting position vector as $2(\overrightarrow{2\text{a}} +\overrightarrow{\text{b}}) - 1 (\overrightarrow{\text{a}} - 2\overrightarrow{\text{b}})$
$= \overrightarrow{3\text{a}} + \overrightarrow{4\text{b}}$
View full question & answer→Question 371 Mark
Find the vector equation of a plane which is at a distance of 5 units from the origin and its normal vector is $2\hat{\text{i}} - 3\hat{\text{j}} + 6\hat{\text{k}}.$
Answer$\overrightarrow{\text{r}}. \frac{(2\hat{\text{i}} - 3\hat{\text{j}} + 6\hat{\text{k})}}{7} = 5$
View full question & answer→Question 381 Mark
$\text{If} \overrightarrow{\text{a}} = 7\hat{\text{i}} + \hat{\text{j}} - 4\hat{\text{k}}$ and $\overrightarrow{\text{b}} = 2\hat{\text{i}} - 6\hat{\text{j}} + 3\hat{\text{k}},$ then find projection of $\overrightarrow{\text{a}}\text{on} \overrightarrow{\text{b}}.$
Answer$\text{p} = \frac{\overrightarrow{a}. \overrightarrow{b}}{\overrightarrow{|b}|} = \frac{8}{7}$
View full question & answer→Question 391 Mark
If vectors $\vec{\text{a}}\ \text{and}\ \vec{\text{b}}$ are such that, $|\vec{\text{a}}|=3,\ |\vec{\text{b}}|=\frac{2}{3}\ \text{and}\ \vec{\text{a}}\times\vec{\text{b}}$ is a unit vector, then write the angle between $\vec{\text{a}}\ \text{and}\ \vec{\text{b}}$.
View full question & answer→Question 401 Mark
Find the projection of the vector $\hat{\text{i}} + 3\hat{\text{j}} + 7\hat{\text{k}}\text{ on the vector } 2\hat{\text{i}} -3\hat{\text{j}} + 6\hat{\text{k}}.$
View full question & answer→Question 411 Mark
The two vectors $\hat{\text{j}} + \hat{\text{k}}$ and $3\hat{\text{i}} - \hat{\text{j}} + 4\hat{\text{k}}$ represent the two sides AB and AC, respectively of a $\triangle \text{ABC},$ Find the length of the median through A.
Answer$\overrightarrow{\text{AD}} = \overrightarrow{\text{AB}} + \frac{1}{2}[\overrightarrow{\text{AC}} - \overrightarrow{\text{AB}}] = \frac{1}{2} (\overrightarrow{\text{AC}} + \overrightarrow{\text{AB}})$
$|\overrightarrow{\text{AD}}| =\frac{1}{2} |3\hat{\text{i}} + 5\hat{\text{k}}| = \frac{1}{2}\sqrt{34}$
View full question & answer→Question 421 Mark
Find the position vector of a point which divides the join of points with position vectors $\overrightarrow{\text{a}} - \overrightarrow{\text{2b }} $ and $\overrightarrow{\text{2a}} +\overrightarrow{\text{b}}$externally in the ratio 2:1.
AnswerGetting position vector as $2(\overrightarrow{2\text{a}} +\overrightarrow{\text{b}}) - 1 (\overrightarrow{\text{a}} - 2\overrightarrow{\text{b}})$
$= \overrightarrow{3\text{a}} + \overrightarrow{4\text{b}}$
View full question & answer→Question 431 Mark
Find the vector equation of a plane which is at a distance of 5 units from the origin and its normal vector is $2\hat{\text{i}} - 3\hat{\text{j}} + 6\hat{\text{k}}.$
Answer$\overrightarrow{\text{r}}. \frac{(2\hat{\text{i}} - 3\hat{\text{j}} + 6\hat{\text{k})}}{7} = 5$
View full question & answer→Question 441 Mark
$\text{If} \overrightarrow{\text{a}} = 7\hat{\text{i}} + \hat{\text{j}} - 4\hat{\text{k}}$ and $\overrightarrow{\text{b}} = 2\hat{\text{i}} - 6\hat{\text{j}} + 3\hat{\text{k}},$ then find projection of $\overrightarrow{\text{a}}\text{on} \overrightarrow{\text{b}}.$
Answer$\text{p} = \frac{\overrightarrow{a}. \overrightarrow{b}}{\overrightarrow{|b}|} = \frac{8}{7}$
View full question & answer→Question 451 Mark
If $\overrightarrow{\text{a}}$and$\overrightarrow{\text{b}}$ are two unit vectors such that $\overrightarrow{\text{a}}$+ $\overrightarrow{\text{b}}$is also a unit vector, then find the angle between $\overrightarrow{\text{a}}$and $\overrightarrow{\text{b}}$.
View full question & answer→Question 461 Mark
Find the projection of the vector $\hat{\text{i}} + 3\hat{\text{j}} + 7\hat{\text{k}}\text{ on the vector } 2\hat{\text{i}} -3\hat{\text{j}} + 6\hat{\text{k}}.$
View full question & answer→Question 471 Mark
If $\overrightarrow{\text{a}} = \text{x}\hat{\text{i}} + 2 \hat{\text{j}} - \text{z}\hat{\text{k}}\text{ and } = \overrightarrow{\text{b}} = 3\hat{\text{i}} - \text{y}\hat{\text{j}} + \hat{\text{k}}$ are two equal vectors, then write the value of x + y + z.
Answer$\because\overrightarrow{\text{a}} = \overrightarrow{\text{b}}$
$\text{x}\hat{\text{i}} + 2\hat{\text{j}} - \text{z}\hat{\text{k}} = 3\hat{\text{i}} - \text{y}\hat{\text{j}} + \hat{\text{k}}$
Equating, we get, x = 3,
$ -\text{y} = 2 \Rightarrow\text{y} = - 2$
$ - \text{z} = 1 \Rightarrow\text{z} = -1$
$\therefore\text{x} + \text{y} +\text{z} = 3-2-1 = 0 .$
View full question & answer→Question 481 Mark
If a unit vector $\overrightarrow{\text{a}}$makes angles $\frac{\pi}{3}$with $\hat{\text{i}},\frac{\pi}{4}$with $\hat{\text{j}}$and an acute angle$\theta$ with $\hat{\text{k}},$ then find the value of $\theta.$
AnswerLet l, m, n, be Direction cosines of $\overrightarrow{\text{a}}$
$\therefore\text{l} =\cos\frac{\pi}{3} =\frac{1}{2};\text{m} = \cos\frac{\pi}{4} = \frac{1}{\sqrt{2}};\text{n} = \cos\theta$
$\because\text{l}^{2} + \text{m}^{2} + \text{n}^{2} = 1 $
$\Rightarrow\bigg(\frac{1}{2}\bigg)^{2} + \bigg(\frac{1}{\sqrt{2}}\bigg)^{2} + \cos^{2}\theta = 1 $
$\Rightarrow\frac{1}{4} + \frac{1}{2} + \cos^{2}\theta = 1 $
$\Rightarrow\cos^{2}\theta = 1 - \bigg(\frac{1}{4} + \frac{1}{2}\bigg) = 1 - \frac{3}{4} = \frac{1}{4}$
$\Rightarrow\cos\theta =\frac{1}{2}\Rightarrow\theta = \frac{\pi}{3}.$
View full question & answer→Question 491 Mark
Find the sum of the vectors $\overrightarrow{a}=\hat{\text{i}}-\hat{\text{2j}}+\hat{\text{k}},\text{ }\overrightarrow{b}=-\hat{\text{2i}}+\hat{\text{4j}}+\hat{\text{5k}}\text{ and }\overrightarrow{c}=\hat{\text{i}}-\hat{\text{6j}}-\hat{\text{7k}}.$
Answer$-\hat{\text{4j}}-\hat{\text{k}}.$
View full question & answer→Question 501 Mark
Find $'\lambda'$ when the projection of $\overrightarrow{a}$ = $\lambda$$\hat{\text{i}}+\hat{\text{j}}+\hat{\text{4k}}$ on $\overrightarrow{b}=\hat{\text{2i}}+\hat{\text{6j}}+\hat{\text{3k}}\text{ is 4 units.}$
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