Sample QuestionsScalar Or Dot Product questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If $\vec{\text{a}}$ and $\vec{\text{b}}$ are unit vectors,then the greatest value of $\sqrt{3}\big|\vec{\text{a}}+\vec{\text{b}}\big|+\big|\vec{\text{a}}-\vec{\text{b}}\big|$ is:
- A
$2$
- B
$2\sqrt{2}$
- ✓
$4$
- D
$\text{None of these}$
Answer: C.
View full solution →If the vectors $\hat{\text{i}}-2\text{x}\hat{\text{j}}+3\text{y}\hat{\text{k}}$ and $\hat{\text{i}}+2\text{x}\hat{\text{j}}-3\text{y}\hat{\text{k}}$ are perpendicular, then the locus of $(x,y)$ is:
Answer: B.
View full solution →Let $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ be three unit vectors, such that $\big|\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}\big|=1$ and $\vec{\text{a}}$ is perpendicular to $\vec{\text{b}}.$ If $\vec{\text{c}}$ makes angle $\alpha$ and $\beta$ with $\vec{\text{a}}$ and $\vec{\text{b}}$ respectively, then $\cos\alpha+\cos\beta=$
- A
$-\frac{3}{2}$
- B
$\frac{3}{2}$
- C
$1$
- ✓
$-1$
Answer: D.
View full solution →What is the length of the longer diagonal of the parallelogram constructed on $5\vec{\text{a}}+2\vec{\text{b}}$ and $\vec{\text{a}}-3\vec{\text{b}}$ if it is given that $|\vec{\text{a}}|=2\sqrt{2},\big|\vec{\text{b}}\big|=3$ and the angle between $\vec{\text{a}}$ and $\vec{\text{b}}$ is $\frac{\pi}{4}$?
- A
$15$
- B
$\sqrt{113}$
- ✓
$\sqrt{593}$
- D
$\sqrt{369}$
Answer: C.
View full solution →If $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ are any three mutualy perpendicular vectors of equal magnitude a, then $\big|\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}\big|$ is equal to
- A
$\text{a}$
- B
$\sqrt{2}\text{a}$
- ✓
$\sqrt{3}\text{a}$
- D
$2\text{a}$
Answer: C.
View full solution →If $|\vec{\text{a}}|=2,\big|\vec{\text{b}}\big|=3$ and $\vec{\text{a}}.\vec{\text{b}}=3,$ find the projection of $\vec{\text{b}}$ on $\vec{\text{a}}.$
View full solution →Find $\vec{\text{a}}.\vec{\text{b}}$ when
$\vec{\text{a}}=\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}$ and $\vec{\text{b}}=4\hat{\text{i}}-4\hat{\text{j}}+7\hat{\text{k}}$
View full solution →Write the projection of the vector $7\hat{\text{i}}+\hat{\text{j}}-4\hat{\text{k}}$ on the vector $2\hat{\text{i}}+6\hat{\text{j}}+3\hat{\text{k}}.$
View full solution →Find $\big|\vec{\text{a}}-\vec{\text{b}}\big|$ if
$|\vec{\text{a}}|=3,\big|\vec{\text{b}}\big|=4$ and $\vec{\text{a}}.\vec{\text{b}}=1$
View full solution →If $\vec{\text{a}}$ and $\vec{\text{b}}$ are two vectors of the same magnitude inclined at an angle of $60^\circ$ such that $\vec{\text{a}}.\vec{\text{b}}=8,$
write the value of their magnitude.
View full solution →If $\vec{\text{a}},\vec{\text{b}}$ are two vectors such that $\big|\vec{\text{a}}+\vec{\text{b}}\big|=\big|\vec{\text{b}}\big|,$ then prove that $\vec{\text{a}}+2\vec{\text{b}}$ is perpendicular to $\vec{\text{a}}.$
View full solution →If $\vec{\text{a}}=2\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}},\vec{\text{b}}=-\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}},$and $\vec{\text{c}}=3\hat{\text{i}}+\hat{\text{j}}$ are such that $\vec{\text{a}}+\lambda\vec{\text{b}}$ is perpendicular to $\vec{\text{c}},$ then find the value of $\lambda.$
View full solution →Find $\lambda$ when the projection of $\vec{\text{a}}=\lambda\hat{\text{i}}+\hat{\text{j}}+4\hat{\text{k}}$ on $\vec{\text{b}}=2\hat{\text{i}}+6\hat{\text{j}}+3\hat{\text{k}}$ is 4 units.
View full solution →Dot product of a vector with vectore $\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}},2\hat{\text{i}}+\hat{\text{j}}-3\hat{\text{k}}$ and $\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ are respectively 4, 0 and 2. Find the vector.
View full solution →Let $\vec{\text{a}}=5\hat{\text{i}}-\hat{\text{j}}+7\hat{\text{k}}$ and $\vec{\text{b}}=\hat{\text{i}}-\hat{\text{j}}+\lambda\hat{\text{k}}.$ Find $\lambda$ such that $\vec{\text{a}}+\vec{\text{b}}$ is orthonal to $\vec{\text{a}}-\vec{\text{b}}.$
View full solution →Prove using vector: the quadrilateral obtained by joining mid-points of adjacent sides of a rectangle is a rhombus.
Express the vector $\vec{\text{a}}=5\hat{\text{i}}-2\hat{\text{j}}+5\hat{\text{k}}$ as the sum of two vectors such that one is parallel to the vector $\vec{\text{b}}=3\hat{\text{i}}+\hat{\text{k}}$ and other is perpendicular to $\vec{\text{b}}.$
View full solution →Express $2\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}}$ as the sum of a vector parallel and a vector perpendicular to $2\hat{\text{i}}+4\hat{\text{j}}-2\hat{\text{k}}.$
View full solution →Show that the points whose position vectors are$\vec{\text{a}}=4\hat{\text{i}}-3\hat{\text{j}}+\hat{\text{k}}, \vec{\text{b}}=2\hat{\text{i}}-4\hat{\text{j}}+5\hat{\text{k}},\vec{\text{c}}=\hat{\text{i}}-\hat{\text{j}}$ from a right triangle.
View full solution →Find the angles which the vector $\vec{\text{a}}=\hat{\text{i}}-\hat{\text{j}}+\sqrt{2}\hat{\text{k}}$ makes with the coordinate axes.
View full solution →