Tag: introduction to vector algebra

Questions Related to introduction to vector algebra

If $ \vec{a} $ and $ \vec{b} $ are two non-collinear unit vectors such that $ |\vec{a}+\vec{b}| = \sqrt{3}, $ find $(2\vec{a}-5\vec{b}).(3\vec{a}+\vec{b}) $ 

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

  1. $ +\dfrac{11}{2} $

  2. $ -\dfrac{13}{2} $

  3. $ -\dfrac{11}{2} $

  4. $ +\dfrac{13}{2} $

Show answer
Correct Option: C
Explanation:

Given $ |\vec{a}+\vec{b}| = \sqrt{3}, $

Now squaring both sides we get,

$(\vec{a}+\vec{b}).(\vec{a}+\vec{b})=3$ [ Since$|\vec{a}|^2=\vec{a}.\vec{a}$ 
or, $|\vec{a}|^2+2\vec{a}.\vec{b}+|\vec{b}|^2=3$ [ Since 

$\vec{a}.\vec{b}=\vec{b}.\vec{a}$ ]
or, $\vec{a}.\vec{b}=\dfrac{1}{2}$.....(1). [ Since $\vec{a},\vec{b}$ are unit vectors then $|\vec{a}|=1=|\vec{b}|$ ]

Now,
$(2\vec{a}-5\vec{b}).(3\vec{a}+\vec{b}) $ 
$=6|\vec{a}|^2-13\vec{a}.\vec{b}-5|\vec{b}|^2$

$=6-\dfrac{13}{2}-5$ [ Using (1)]
$=-\dfrac{11}{2}$.

Which of the following can represent a vector?

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

  1. The length of the distance between the points $(0,0)$ and $(2,7)$

  2. A line segment beginning at $(2,7))$ and ending at $(0,0)$

  3. The length of the distance between the points $(2,7)$ and $(0,0)$

  4. A line segment beginning at $(0,0)$ and ending at $(2,7)$

Show answer
Correct Option: B,D
Explanation:
A vector is a quantity that can be described as having both magnitude and direction.
The length of the distance between any two points is a magnitude with no direction, so it can't represent a vector.
A line segment beginning at a certain point and ending at another can represent a vector. The magnitude of the vector is the distance between the points, and its direction is the direction from the initial point to the terminal point.
The following can represent a vector:
A line segment beginning at $(0,0)$ and ending at $(2,7)$.
A line segment beginning at $(2,7)$ and ending at $(0,0)$

Direction of zero vector

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

  1. does not exist

  2. towards origin

  3. indeterminate

  4. None of these

Show answer
Correct Option: C
Explanation:

As, zero vector represents a point.
Direction is indeterminate.
Hence, option C.

Which will result in a vector?

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

  1. Product of a scalar and a scalar.

  2. Product of a scalar and a vector.

  3. Addition of two vectors

  4. None of these

Show answer
Correct Option: B,C
Explanation:
Let two vectors
$\vec{a}=\hat{i}$ 
$\vec{b}=\hat{i}+\hat{j}$
Addition of both vector 
$\vec{a}+\vec{b}=\hat{i}+\hat{i}+\hat{j}$
$\vec{a}+\vec{b}=2\hat{i}+\hat{j}$
Here we get vector by addition of both vectors 
hence option C is correct

let two scalar $\lambda=2,\mu=1$
$\lambda\times\mu=2\times1=2$
SO from here we get a scalar quantity Hence 
Option A is not correct 

$\lambda\times\vec{a}=\lambda\hat{i}$
Here vector quantity is obtained 
hence option B is correct

What is the value of $p$ for which the vector $p\left( 2\hat { i } -\hat { j } +2\hat { k }  \right)$ is of $ 3$ units length?

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

  1. $1$

  2. $2$

  3. $3$

  4. $6$

Show answer
Correct Option: A
Explanation:

length of vector $a\hat { i } +b\hat { j } +c\hat { k } $ from origin is $\sqrt { a^2+b^2+c^2 } $ 

So $\sqrt { {(2p)}^2+{(-p)}^2+{(2p)}^2 } =\sqrt { 9p^2 }=3p $
Length is $3$ units given. 
$\therefore 3p=3\implies p=1$
Hence, A is correct.

If $\vec{x}$ and $\vec{y}$ be unit vectors and $\displaystyle |\vec{z}| = \dfrac{2}{\sqrt 7}$ such that $\vec{z} + (\vec{z} \times \vec{x}) = \vec{y}$ and $\theta$ is the angle between $\vec{x}$ and $\vec{z}$, then the value of sin $\theta$ is

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

  1. $\displaystyle \dfrac{1}{2}$

  2. $1$

  3. $\displaystyle \dfrac{\sqrt 3}{2}$

  4. $\displaystyle \dfrac{\sqrt 3 -1}{2 \sqrt 2}$

Show answer
Correct Option: C
Explanation:

$|\vec{z} + (\vec{z} \times \vec{x}) | = | \vec{y}|^2 \,\,\,\,\,\Rightarrow$
$|\vec{z}|^2+|\vec{z}|^2 |\vec{x}|^2 \,sin^2\,\theta =1$
$\displaystyle \Rightarrow \,|z| = \frac{1}{\sqrt {1 + sin^2\,\theta}} = \frac{2}{\sqrt 7} \Rightarrow sin\,\theta = \frac{\sqrt 3}{2}$