Tag: operations on vectors

Questions Related to operations on vectors

In a square matrix $A$ of order $3$ the elements, $a _{i\ i}s^{'}$ are the sum of the roots of the equation $x^{2}-(a+b)x+ab=0;\ a _{1,\ i+1}s^{'}$ are the product of the roots, $a _{1,\ i-i}s^{'}$ are all unity and the rest of the elements all zero. The value of the det. $(A)$ is equal to

maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. $0$

  2. $(a+b)^{3}$

  3. $a^{3}-b^{3}$

  4. $(a^{2}+b^{2})$

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Correct Option: B
Explanation:
Given, square matrix $A$ of order $3$ with $a _{ii}= (a+b) $ (sum of roots)

$a _{1 i+1}=a _{12}=ab$  (product of roots)

$a _{1i-1}= a _{10}=1$

$|A|= \begin{vmatrix} (a+b) & 0 & 0 \\ 1 & (a+b) & ab \\ 0 & 0 & (a+b) \end{vmatrix}= (a+b)[(a+b)(a+b)-0]$

$=(a+b)^{3}$

$\therefore $ Option $B$ is correct.


If $(\vec{a}\times\vec{b})^{2}+(\vec{a}.\vec{b})^{2}=144$ and $|\vec{a}|=4,\ |\vec{b}|=$

maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. $16$

  2. $8$

  3. $3$

  4. $5$

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Correct Option: C

One of the following is not a vector

maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. displacement

  2. work

  3. centrifugal force

  4. gravitational field

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Correct Option: B
Explanation:

Displacement,Centrifugal force and Gravitation are vectors.
Work $W=\overrightarrow { F } \cdot \overrightarrow { S } $ a scalar.
Hence, option B.

Which one of the following is not a scalar quantity?

maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. Temperature

  2. Density

  3. Mass

  4. Weight

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Correct Option: D
Explanation:

Temperature,Density,Mass are clearly scalars.
Whereas in case of weight, it acts always downwards beacuse of
accelration due to gravity.
Hence, option D.

If a be an unit vector then

maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. direction of a is constant

  2. magnitude of a is constant

  3. direction and magnitude of a is constant

  4. any one of direction or magnitude is constant.

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Correct Option: C
Explanation:

For a given unit vector , Magnitude and direction are constant (fixed).
since, a vector cannot have two directions.
Hence, option C.

If two vectors have the same magnitude and direction regardless of the positions of their initial points, then they are

maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. Co-initial vectors

  2. Colinear vectors

  3. Equal Vectors

  4. Unit Vector

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Correct Option: C
Explanation:

 A quantity that has magnitude as well as direction is called a vector.  

or we can say ,
Two vectors are said to be equal if they have the same magnitude and direction, regardless of the positions of their initial points.

A line PQ is symbolically written as

maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. PQ

  2. $\overrightarrow{PQ}$

  3. $\overleftrightarrow{PQ}$

  4. None of these

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Correct Option: C
Explanation:

We know that the symbolically is  $\leftrightarrow$

now,  option $C$ is correct.

A ray has ................ end point/points.

maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. 1

  2. 2

  3. 3

  4. no

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Correct Option: A
Explanation:

In geometry, a ray is usually taken as a half-infinite line (also known as a half-line) with one of the two points and taken to be at infinity. 

Which of the following symbols represent vectors?

maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. $v$

  2. $\vec{AB}$

  3. $\displaystyle \left| v \right| $

  4. $\displaystyle u+\left| v \right| $

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Correct Option: B
Explanation:

The vector is visual representation of a physical quantity that which has the both direction and the magnitude.

And out of these option,
Only option $B$ have the both direction and magnitude.
So, the correct answer is $\vec {AB}$.
Because it has a particular direction from $A$ to $B$ and also have the magnitude.

Which of the following expressions represent vectors ?

maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. $v$

  2. $AB$

  3. $\displaystyle \left| v \right| $

  4. $\displaystyle \vec { AB }  $

Show answer
Correct Option: D
Explanation:

The vector is visual representation of a physical quantity that which has the both direction and the magnitude.

And out of these option,
Only option $D$ have the both direction and magnitude.
So, the correct answer is $\vec {AB}$.
Because it has a particular direction from $A$ to $B$ and also have the magnitude.