Mathematics · Physics

Vector Algebra and Calculus

192 Questions

Vector algebra involves mathematical operations on spatial quantities including dot products and cross products. These questions test the understanding of vector spaces and linear combinations. This topic is crucial for advanced mathematics and physics exams.

Dot and cross productsVector linear combinationsPerpendicular vector calculationsVector space dimensionsCollinear points and vectors

Vector Algebra and Calculus Questions

Multiple choice

The magnitude of the resultant of the two vectors is always _____________

  1. Greater than one of the vector’s magnitude

  2. Smaller than one of the vector’s magnitude

  3. Depends on the angle between them

  4. Axis we choose to calculate the magnitude
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Yes, the magnitude of the resultant of the two vectors always depends on the angle between them. It might be greater or smaller than one of the vector’s length. For perfectly saying, it does depends upon the angle between them.

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

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

  1. Co-initial vectors

  2. Colinear vectors

  3. Equal Vectors

  4. Unit Vector
Reveal answer Fill a bubble to check yourself
C Correct answer
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.

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

Which of the following symbols represent vectors?

  1. $v$
  2. $\vec{AB}$
  3. $\displaystyle \left| v \right| $
  4. $\displaystyle u+\left| v \right| $
Reveal answer Fill a bubble to check yourself
B Correct answer
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.

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

Which of the following expressions represent vectors ?

  1. $v$
  2. $AB$
  3. $\displaystyle \left| v \right| $
  4. $\displaystyle \vec { AB } $
Reveal answer Fill a bubble to check yourself
D Correct answer
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.

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

Let        $\dot{a}$ = $\hat{i}$ + $\hat{j}$ + $\sqrt{2}\hat{k}$
              $\dot{b}$ = $b _1\hat{i}$ + $b _2\hat{j}$ + $\sqrt{2}\hat{k}$
              $\dot{c}$ = $5\hat{i}$ + $\hat{j}$ + $\sqrt{2}\hat{k}$
& ($\dot{a}$ + $\dot{b}$) is perpendicular to \overrightarrow{c} and projection vector of $\dot{b}$ on $\overrightarrow{a}$ is $\overrightarrow{a}$ then find $\left | \overrightarrow{b} \right |$

  1. 6
  2. $\sqrt{22}$
  3. $\sqrt{32}$

  4. 11
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given the conditions that (a+b) is perpendicular to c and the projection of b on a is a, one can solve for the components of b. The calculation leads to |b| = sqrt(22).

Multiple choice maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors 
Let a=i+j+k and c=j-k . If b is a vector satisfying a×b=c and a.b=3, then find b.
  1. $\dfrac{1}{3}(5\hat{i}+2\hat{j}+2\hat{k})$
  2. $\dfrac{1}{3}(2\hat{i}+3\hat{j}+\hat{k}$
  3. $\displaystyle 2\overrightarrow{a}$
  4. $\displaystyle -2\overrightarrow{a}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let$b=xi+yj+zk$

Now 
$a.b=x+y+z=3$
Now $a\times b=(i+j+k)(xi+yj+zk)\ \quad=(z-y)i+(x-z)j+(y-x)k=j-k$
Now $z.y=0$ hence $k=y$
Now  $x-z=1$ and $y-x=-1$
Now
$x+y+z=3\1+z+z+z=3\z=\cfrac{2}{3}$
Hence $y=\cfrac{2}{3}\x=\cfrac{5}{3}$
Now $\overrightarrow{b}=\cfrac{(5i+2j+2k)}{3}$