12th Standard CBSE Syllabus & Materials
12th Standard CBSE
CBSE 12th Economics Government Budget and the Economy Previous year Question Papers Study Material - QB365 Set A
NEW12th Standard CBSE
CBSE 12th Computer Science Interface Python with MySQL - New Previous year Question Papers Study Material - QB365 Set A
NEW12th Standard CBSE
CBSE 12th Computer Science Database Concept - New Previous year Question Papers Study Material - QB365 Set A
NEW12th Standard CBSE
CBSE 12th Computer Science Data Communication - New Previous year Question Papers Study Material - QB365 Set A
NEW12th Standard CBSE
CBSE 12th Computer Science Data Structures - New Previous year Question Papers Study Material - QB365 Set A
NEW12th Standard CBSE
CBSE 12th Computer Science Functions - New Previous year Question Papers Study Material - QB365 Set A
Published on: 29/08/2019
Vector Algebra
Download CBSE Class 12th Standard CBSE Maths question papers, sample papers, important questions, and previous year solved papers in PDF format. Get free study materials, NCERT solutions, and exam preparation resources for Class 12th Standard CBSE Maths
Questions + Answers key
Take MCQ Maths Test
1.
Write the value of p for which \(\overrightarrow { a } =3\hat { i } +2\hat { j } +9\hat { k } \quad and\quad \overrightarrow { b } =\hat { i } +p\hat { j } +3\hat { k } \) are parallel vectors.
2.
Find the projection of \(\overrightarrow { a } \ on\ \overrightarrow { b } \) if \(\overrightarrow { a } .\overrightarrow { b } =8\ and\ \overrightarrow { b } =2\overrightarrow { i } +6\overrightarrow { j } +3\overrightarrow { k } \)
3.
Find a vector in the direction of \(\overrightarrow { a } =\overrightarrow { i } -2\overrightarrow { j } \) whose magnitude is 7.
4.
If \(\left| \overrightarrow { a } \right| =\sqrt { 3 } ,\left| \overrightarrow { b } \right| =2\) and angle between \(\overrightarrow { a } \ and\ \overrightarrow { b } \) is 60o find \(\overrightarrow { a } .\overrightarrow { b } \)
5.
If \(\overrightarrow { a } =\hat { i } +2\hat { j } -3\hat { k } \) and \(\overrightarrow { b } =2\hat { i } +4\hat { j } +9\hat { k } \) find a unit vector parallel to \(\overrightarrow { a } +\overrightarrow { b } \)
6.
If \(\left| \overrightarrow { a } \right| =2,\left| \overrightarrow { b } \right| =\sqrt { 3 } \) and \(\overrightarrow { a } .\overrightarrow { b } =\sqrt { 3 } \) find the angle between \(\overrightarrow { a } \ and\ \overrightarrow { b } \)
7.
If \(\overrightarrow { a } =\overrightarrow { i } +2\overrightarrow { j } -\overrightarrow { k } \ and\ \overrightarrow { b } =3\overrightarrow { i } +\overrightarrow { j } -5\overrightarrow { k } \) find a unit vector in the direction of \(\overrightarrow { a } -\overrightarrow { b } \)
8.
For what value of \(\lambda\) are the vectors \(\overrightarrow { a } =2\overrightarrow { i } +\lambda \overrightarrow { j } +\overrightarrow { k } \) and \(\overrightarrow { b } =\overrightarrow { i } -2\overrightarrow { j } +3\overrightarrow { k } \) perpendicular to each other?
9.
Find a unit vector in the direction of \(\overrightarrow { a } =3\overrightarrow { i } -2\overrightarrow { j } +6\overrightarrow { k } \)
10.
For any two vectors \(\vec{a} \text { and } \vec{b}\)
\(\vec{a}+\vec{b}=\vec{b}+\vec{a}\) (Commutative property)
1.
\(\frac{3}{1}=\frac{2}{p}=\frac{9}{3} \Rightarrow p=\frac{2}{3}\)
2.
\(\text { Projection of } \vec{a} \text { on } \vec{b}=\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}=\frac{8}{7}\)
3.
Vector of magnitude 7 along is \(7 \hat{a}=\frac{7(\hat{i}-2 \hat{j})}{\sqrt{5}}\)
\(7\hat { a } =\frac { 7 }{ \sqrt { 5 } } \hat { i } -\frac { 14 }{ \sqrt { 5 } } \hat { j } \)
4.
\(\cos 60^{\circ}=\frac{\vec{a} \cdot \vec{b}}{2 \sqrt{3}} \Rightarrow \frac{1}{2}=\frac{\vec{a} \cdot \vec{b}}{2 \sqrt{3}} \Rightarrow \vec{a} \cdot \vec{b}=\sqrt{3}\)
5.
\(\frac { 1 }{ 3 } \hat { i } +\frac { 2 }{ 3 } \hat { j } +\frac { 2 }{ 3 } \hat { k } \)
6.
\(\cos \theta=\frac{\vec{a} \cdot \vec{b}}{|\vec{a} \| \vec{b}|}=\frac{\sqrt{3}}{2 \sqrt{3}}=\frac{1}{2} \Rightarrow \theta=\frac{\pi}{3}\)
7.
\(\vec{r}=\vec{a}-\vec{b}=-2 \hat{i}+\hat{j}+4 \hat{k},|\vec{r}|=\sqrt{4+1+16}=\sqrt{21} \)
\(\text { Now, unit vector } \hat{r}=\frac{\vec{r}}{|\vec{r}|}=\frac{-2}{\sqrt{21}} \hat{i}+\frac{1}{\sqrt{21}} \hat{j}+\frac{4}{\sqrt{21}} \hat{k} \text { ; } \)
8.
\(\text { If } a \text { and } \vec{b} \text { are perpendicular, then } \vec{a} \cdot \vec{b}=0\)
\(\Rightarrow 2-2 \lambda+3=0 \Rightarrow \lambda=\frac{5}{2}\)
9.
\(\hat{a}=\frac{\vec{a}}{|\vec{a}|}=\frac{3 \hat{i}-2 \hat{j}+6 \hat{k}}{\sqrt{9+4+36}}=\frac{3}{7} \vec{i}-\frac{2}{7} \vec{j}+\frac{6}{7} \vec{k}\)
10.
Consider the parallelogram ABCD. Let \(\overrightarrow{\mathrm{AB}}=\vec{a} \text { and } \overrightarrow{\mathrm{BC}}=\vec{b}\) then using the triangle law, from triangle ABC, we have \(\overrightarrow{\mathrm{AC}}=\vec{a}+\vec{b}\)
Now, since the opposite sides of a parallelogram are equal and parallel, from figure \(\overrightarrow{\mathrm{AD}}=\overrightarrow{\mathrm{BC}}=\vec{b}\) \(\overrightarrow{\mathrm{DC}}=\overrightarrow{\mathrm{AB}}=\vec{a} .\)
Again using triangle law, from triangle ADC, we have
\( \overrightarrow{\mathrm{AC}}=\overrightarrow{\mathrm{AD}}+\overrightarrow{\mathrm{DC}}=\vec{b}+\vec{a} \\ \vec{a}+\vec{b}=\vec{b}+\vec{a} \)
12th Standard CBSE Syllabus & Materials
12th Standard CBSE
CBSE 12th Computer Science Python Revision Tour I - New Previous year Question Papers Study Material - QB365 Set A
NEW12th Standard CBSE
CBSE 12th Business Studies Planning Important Questions And Answers Study Material - QB365 Set A
NEW12th Standard CBSE
CBSE 12th Business Studies Business Environment Important Questions And Answers Study Material - QB365 Set A
NEW12th Standard CBSE
CBSE 12th Business Studies Principles of Management Important Questions And Answers Study Material - QB365 Set A
CBSE 12th Standard CBSE Subjects
CBSE Standards
