Product of Two Vectors: Overview, Questions, Preparation

The scalar product of two non-zero vectors $\vec{a}$  and $\vec{b}$ is denoted by $\vec{a}$ . $\stackrel{\to }{b}$ (read as $\vec{a}$  dot $\vec{b}$ ) is defined as

. $\vec{a}$ $\stackrel{\to }{b}$ = $\left|\stackrel{\to }{a}\right|$ $\left|\stackrel{\to }{b}\right|$ cos θ

Where θ is the angle between $\vec{a}$ and $\vec{b}$ and 0 $\le$  θ $\le$ π.

If either $\vec{a}$ = $\stackrel{\to }{0}$ or $\vec{b}$ = $\stackrel{\to }{0}$ , then the angle θ between $\vec{a}$  and $\vec{b}$ is not defined and in this case, we define $\vec{a}$ . $\vec{b}$ = 0.

$\vec{a}$ . $\vec{b}$ is always a scalar. That is why, dot product is called scalar product.

$\vec{a}$ . $\stackrel{\to }{b}$ can be positive, negative or zero, according as cos θ is positive, negative or zero.

If $\stackrel{\to }{b}$ and $\stackrel{\to }{b}$  are like vectors (i.e., θ = 0), then $\stackrel{\to }{a}$  . $\stackrel{\to }{b}$ = $\left|\stackrel{\to }{a}\right|$ $\left|\stackrel{\to }{b}\right|$ cos 0 = $\left|\stackrel{\to }{a}\right|$ $\left|\stackrel{\to }{b}\right|$ .

If $\vec{a}$ and $\vec{b}$  are unlike vectors (i.e., θ = π ), then $\stackrel{\to }{a}$ . $\vec{b}$  = $\left|\stackrel{\to }{a}\right|$ $\left|\stackrel{\to }{b}\right|$ cos π = – $\left|\stackrel{\to }{a}\right|$ $\left|\stackrel{\to }{b}\right|$ .

Square of vector $\vec{a}$ is given by ${\stackrel{\to }{a}}^2$ = $\vec{a}$ . $\vec{a}$ = $\left|\stackrel{\to }{a}\right|$ $\left|\stackrel{\to }{a}\right|$ cos 0 = ${\left|\stackrel{\to }{a}\right|}^2$ .

Scalar product is commutative, i.e., $\vec{a}$ $\vec{b}$ .= . $\vec{b}$ . $\vec{a}$ $\hat{k}$

Distributivity of scalar product over addition: Let $\stackrel{\to }{a}$ , $\stackrel{\to }{b}$ and $\stackrel{\to }{c}$ be any three vectors, then $\vec{a}$

.( $\stackrel{\to }{c}$ + $\stackrel{\to }{c}$ ) = $\vec{a}$ . $\vec{b}$  + $\vec{a}$ . $\stackrel{\to }{c}$

Let $\vec{a}$ and $\vec{b}$  be any two vectors and let $\lambda$  be any scalar, then

$\lambda$   ( $\vec{a}$ . $\vec{b}$ ) = ( $\lambda$ $\vec{a}$ ). $\vec{b}$ = $\vec{a}$ .( $\lambda$ $\vec{b}$ ).

Let $\vec{a}$ = $a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ and $\vec{b}$ = $b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ , then

$\vec{a}$ . $\vec{b}$ = ( $a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ ) . ( $b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ ) = $a_1{b}_1+a_2{b}_2+a_3{b}_3$ .

For unit vectors $\hat{i}$ , $\hat{j}$ , $\hat{k}$ , we have

  $\hat{i}$ $\hat{i}$    . =1 $\hat{j}$ $\hat{j}$ =1 $\hat{k}$ $\hat{k}$ = 1

  $\hat{i}$ $\hat{j}$   . = 0 $\hat{j}$ $\hat{k}$   = 0 $\hat{k}$ $\hat{i}$ = 0

$\hat{j}$ . $\hat{i}$ = 0 $\hat{k}$   $\hat{j}$ = 0 $\hat{i}$ $\hat{k}$   = 0

Let $\vec{a}$ and $\vec{b}$ be two non-zero vectors. Then, vector product (or cross product) of $\vec{a}$  and $\vec{b}$  is denoted by $\vec{a}$ $\times$ $\vec{b}$  (read as $\stackrel{\to }{a}$ ‘ cross $\vec{b}$ ’) and is defined as  $\vec{a}$ $\times$ = $\left|\stackrel{\to }{a}\right|$ $\left|\stackrel{\to }{b}\right|$ sin θ $\hat{n}$

Where θ is the angle between $\vec{a}$ and $\vec{b}$ , 0 $\le$  θ $\le$ π and $\hat{n}$ is a unit vector perpendicular to the plane containing the vectors $\vec{a}$  and $\vec{b}$ , which is given by ‘ Right-hand Thumb Rule’.

According to this rule, if we bend the fingers of the right hand in such a way that they point in the direction of rotation from $\vec{b}$ to $\vec{b}$  through the smaller angle between them, then the thumb points in the direction of vector $\vec{a}$ $\times$ $\vec{b}$ .

If either $\vec{a}$ = $\stackrel{\to }{0}$ or $\vec{b}$ = $\stackrel{\to }{0}$ , then the angle θ between $\vec{a}$  and $\vec{b}$  is not defined and in this case, we define $\vec{a}$ $\times$ = $\stackrel{\to }{0}$ .

$\vec{a}$ $\times$ $\vec{b}$ is always a vector. That is why, cross product is called vector product.

If $\vec{a}$ and $\vec{b}$ are like vectors (i.e., θ = 0), then $\left|\stackrel{\to }{a}\times \stackrel{\to }{b}\right|$  = $\left|\stackrel{\to }{a}\right|$ $\left|\stackrel{\to }{b}\right|$ sin 0 = 0 and $\vec{a}$ $\times$ $\vec{b}$ = $\stackrel{\to }{0}$ .

If $\vec{a}$ and $\vec{b}$  are unlike vectors (i.e., θ = π ), then $\left|\stackrel{\to }{a}\times \stackrel{\to }{b}\right|$ = sin π = 0 and $\vec{a}$ $\times$ $\vec{b}$  = $\stackrel{\to }{0}$ .

Vector product is not commutative, i.e. $\vec{a}$ $\times$ $\vec{b}$ , ≠ $\vec{b}$ $\times$ $\vec{a}$ .

$\left|\stackrel{\to }{a}\times \stackrel{\to }{b}\right|$ = $\left|\stackrel{\to }{a}\right|$ $\left|\stackrel{\to }{b}\right|$ sin θ = $\left|\stackrel{\to }{b}\times \stackrel{\to }{a}\right|$ and direction of  $\vec{a}$ $\times$ $\stackrel{\to }{b}$ is opposite to that of $\stackrel{\to }{b}$ $\times$ $\vec{a}$ .

Thus, $\vec{a}$ $\times$ $\vec{b}$ = – $\vec{b}$ $\times$ $\vec{a}$ .

Distributivity of Vector Product over Addition: Let $\vec{a}$ $\vec{b}$ ,  and $\stackrel{\to }{c}$ be any three vectors,    then $\vec{a}$ $\times$ ( $\vec{b}$ + $\stackrel{\to }{c}$ ) = $\vec{a}$ $\times$ $\vec{b}$ + $\vec{a}$ $\times$ $\stackrel{\to }{c}$ .

Let $\vec{a}$ and $\stackrel{\to }{b}$ be any two vectors and let $\lambda$ be any scalar, then

$\lambda$ ( $\vec{a}$ $\times$ $\vec{b}$ ) = ( $\lambda$ $\vec{a}$ ) $\times$ $\vec{b}$  = $\vec{a}$ $\times$  ( $\lambda$ $\vec{b}$ ).

For unit vectors $\hat{i}$ , $\hat{j}$ , $\hat{k}$ , we have

$\hat{i}$ $\times$ $\hat{i}$ = $\stackrel{\to }{0}$ $\hat{j}$ $\times$ $\hat{j}$                           =       $\stackrel{\to }{0}$ $\hat{k}$ $\hat{k}$                   = $\stackrel{\to }{0}$

$\hat{i}$ $\times$ $\hat{j}$ = $\hat{k}$ $\hat{j}$ $\times$ $\hat{k}$                          =   $\hat{i}$ $\hat{k}$ $\times$ $\hat{i}$                        = $\hat{j}$

$\hat{j}$ $\times$ $\hat{i}$   = – $\hat{k}$ $\hat{k}$ $\times$ $\hat{j}$                       = – $\hat{i}$ $\hat{i}$ $\times$ $\hat{k}$                        = – $\hat{j}$ .

Let $\vec{a}$  = $a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ and $\stackrel{\to }{b}$ = $b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ , then

$\vec{a}$ $\times$ $\stackrel{\to }{b}$ = ( $a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ ) $\times$ ( $b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ )

  = $a_1\hat{i}$ $\times$ ( $b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ ) + $a_2\hat{j}$ $\times$ ( $b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ ) + $a_3\hat{k}$ $\times$ ( $b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ )

= 0 $a_1{b}_2$ $\hat{k}$ – $a_1{b}_3$ $\hat{j}$ – $a_2{b}_1$ $\hat{k}$ + 0 + $a_2{b}_3$ $\hat{i}$ + $a_3{b}_1$ $\hat{j}$ – $a_3{b}_2$ $\hat{i}$ + 0

= $\hat{i}$ ( $a_2{b}_3$ – $a_3{b}_2$ ) – $\hat{j}$ ( $a_1{b}_3$ – $a_3{b}_1$ ) + $\hat{k}$ ( $a_1{b}_2$ – $a_2{b}_1$ )

= $\left|\begin{array}{ccc}\hfill \hat{i}\hfill & \hfill \hat{j}\hfill & \hfill \hat{k}\hfill \\ \hfill a_1\hfill & \hfill a_2\hfill & \hfill a_3\hfill \\ \hfill b_1\hfill & \hfill b_2\hfill & \hfill b_3\hfill \end{array}\right|$

Dot product: Result is a scalar, involves cosine of the angle. 

Cross product: Result is a vector, involves sine of the angle, and a direction perpendicular to both input vectors. 

1)  Dot Product (Scalar Product)

a) The dot product of two vectors, denoted by a ⋅ b, is a scalar quantity calculated by multiplying the magnitudes of the two vectors and the cosine of the angle between them.

b) Formula: a ⋅ b = |a| |b| cos θ, where |a| and |b| are the magnitudes of vectors a and b, and θ is the angle between them.

c) Result: The dot product is a scalar (a single number). 

2) Cross Product (Vector Product):

a) The cross product of two vectors, denoted by a × b, is a vector quantity that is perpendicular to both of the original vectors. 

b) Formula: a × b = |a| |b| sin θ n̂, where |a| and |b| are the magnitudes of vectors a and b, θ is the angle between them, and n̂ is a unit vector perpendicular to both a and b. 

c) Direction: The direction of the resulting vector is determined by the right-hand rule. 

d) Result: The cross product is a vector. 

3) $\vec{a}$ . $\vec{b}$  is always a scalar. That is why, dot product is called scalar product.

$\vec{a}$ $\stackrel{\to }{b}$ can be positive, negative or zero, according as cos θ is positive, negative or zero.

4) If $\vec{a}$ and $\vec{b}$ are like vectors (i.e., θ = 0), then $\stackrel{\to }{a}$ . $\vec{b}$  = $\left|\stackrel{\to }{a}\right|$ $\left|\stackrel{\to }{b}\right|$ cos 0 = $\left|\stackrel{\to }{a}\right|$ $\left|\stackrel{\to }{b}\right|$ .

5) If $\vec{a}$ and $\vec{b}$  are unlike vectors (i.e., θ = π ), then $\stackrel{\to }{a}$ $\vec{b}$ .  = $\left|\stackrel{\to }{a}\right|$ $\left|\stackrel{\to }{b}\right|$ cos π = – $\left|\stackrel{\to }{a}\right|$ $\left|\stackrel{\to }{b}\right|$ .

6) Square of vector $\vec{a}$  is given by ${\stackrel{\to }{a}}^2$ = $\vec{a}$ . $\vec{a}$ = $\left|\stackrel{\to }{a}\right|$ $\left|\stackrel{\to }{a}\right|$ cos 0 = ${\left|\stackrel{\to }{a}\right|}^2$ .

7) Scalar product is commutative, i.e., $\vec{a}$ $\vec{b}$ .  = . $\vec{b}$ $\vec{a}$ .

$8)\vec{a}$ $\times$ $\vec{b}$ is always a vector. That is why, cross product is called vector product.

8) If $\vec{a}$ and $\vec{b}$ are like vectors (i.e., θ = 0), then $\left|\stackrel{\to }{a}\times \stackrel{\to }{b}\right|$  = $\left|\stackrel{\to }{a}\right|$ $\left|\stackrel{\to }{b}\right|$ sin 0 = 0 and $\vec{a}$ $\times$ $\vec{b}$  = $\stackrel{\to }{0}$ .

9) If $\vec{a}$  and $\vec{b}$ are unlike vectors (i.e., θ = π ), then $\left|\stackrel{\to }{a}\times \stackrel{\to }{b}\right|$ = $\left|\stackrel{\to }{a}\right|$ $\left|\stackrel{\to }{b}\right|$ sin π = 0 and $\vec{a}$ $\times$ $\vec{b}$ = . $\stackrel{\to }{0}$

10)  Vector product is not commutative $\vec{a}$ $\times$ $\vec{b}$ , i.e., ≠ $\vec{b}$ $\times$ $\vec{a}$

.

$\left|\stackrel{\to }{a}\times \stackrel{\to }{b}\right|$ = $\left|\stackrel{\to }{a}\right|$ $\left|\stackrel{\to }{b}\right|$ sin θ = $\left|\stackrel{\to }{b}\times \stackrel{\to }{a}\right|$ and direction of  $\vec{a}$ $\times$ $\stackrel{\to }{b}$ is opposite to that of $\stackrel{\to }{b}$ $\times$ $\vec{a}$ .

Thus, $\vec{a}$ $\times$ $\vec{b}$ = – $\vec{b}$ $\times$ $\vec{a}$ .