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

$A$ vector $\vec V$ is inclined at equal angles to axes $OX,OY$ and $OZ$. If $\vec V$ is $6units$, then $\vec V$ is

  1. $2\sqrt 3\left( \hat i+\hat j+\hat k \right )$
  2. $2\sqrt 3\left( \hat i-\hat j+\hat k \right )$
  3. $\sqrt 2\left( \hat i+\hat j+\hat k \right )$
  4. $2\sqrt 3\left( \hat i+\hat j-\hat k \right )$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

If a vector of magnitude 6 is inclined at equal angles to the axes, its components are equal (x=y=z). Thus, V = k(i + j + k). Since |V| = 6, k * sqrt(3) = 6, so k = 6/sqrt(3) = 2 * sqrt(3).

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

$\sum _{ i=1 }^{ n }{ \vec { ai }  } =\vec { 0 } \quad where\quad |\vec { a\quad i\quad | } =1\forall i$ then the value of $\sum _{ 1\le i }^{  }{ \sum _{ <j\le n }^{  }{ \vec { { a } _{ i } }  }  } .\vec { { a } _{ j } } $ is 

  1. -n/2

  2. -n

  3. n/2

  4. n
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given sum(ai) = 0, we have |sum(ai)|^2 = 0. Expanding this, sum(|ai|^2) + 2 * sum(ai . aj) = 0. Since |ai| = 1, sum(1) + 2 * sum(ai . aj) = 0, so n + 2 * sum(ai . aj) = 0, which gives sum(ai . aj) = -n/2.

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

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}) $ 

  1. $ +\dfrac{11}{2} $
  2. $ -\dfrac{13}{2} $
  3. $ -\dfrac{11}{2} $
  4. $ +\dfrac{13}{2} $
Reveal answer Fill a bubble to check yourself
C Correct answer
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)]
$=-\dfrac{11}{2}$.

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

Which of the following can represent a vector?

  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 $(2,7)$
Reveal answer Fill a bubble to check yourself
B,D Correct answer
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 $(0,0)$
Multiple choice maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

Which will result in a vector?

  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
Reveal answer Fill a bubble to check yourself
B,C Correct answer
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
Multiple choice maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

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?

  1. $1$
  2. $2$
  3. $3$
  4. $6$
Reveal answer Fill a bubble to check yourself
A Correct answer
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.

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

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

  1. $\displaystyle \dfrac{1}{2}$
  2. $1$
  3. $\displaystyle \dfrac{\sqrt 3}{2}$
  4. $\displaystyle \dfrac{\sqrt 3 -1}{2 \sqrt 2}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

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

Multiple choice maths rule of three inverse proportion direct proportion and inverse proportion types of proportions

A car takes 2 hours to reach a destination by travelling at the speed of 60 km/h. when the car travels at the speed of 80 km/h it takes 

  1. one and half hour

  2. one hour

  3. two hour

  4. None
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let the car takes x hours to reach a destination by travelling at the speed of 80 km/h. Then,

Speed (in km/hr) 60 80
Time (in hours) 2 x

Clearly, more the speed, less will be the time taken. So, it is a case of inverse proportion.
$ \therefore \,\,\, 60 \times 2 = 80 \times x  \Rightarrow x = \displaystyle \frac{60 \times 2}{80} = \frac{3}{2}$
Hence, the time taken will be $\displaystyle 1\frac{1}{2}$ hours.

Multiple choice maths rule of three inverse proportion direct proportion and inverse proportion types of proportions

Which is an example of inverse proportion?

  1. More amount of sweets, more total cost

  2. More length of cloth, more cost

  3. More expenditure, less saving

  4. More height of object, more length of its shadow
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

2 quantities say it be $x , y$ are said to be in proportion when the change in value of $x$ , leads to the equal change in value of $y$.

If $x$ increases and hence  $y$ decreases proportionally, it is called Inverse proportion i.e $x\propto\dfrac{1}{y}$
For example, In option C, if our expenses are more, then we are left with lesser savings from a fixed salary or income.
hence, Expenditure and savings are in inverse proportion.
So the answer is C