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Scalar

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In linear algebra, a real number is considered as scalar, and we can also relate the term scalar with the vectors in vector space by using the operation known as Scalar Multiplication. In this operation a scalar or a number is multiplied by a vector to get back another vector.

Scalar is a physical quantity which has only magnitude and has no direction is called a scalar quantity or a scalar.

Example: Mass, length, volume, speed, temperature etc., are scalar quantities. This means giving the number specifies how much of it is there.

If $\vec{a}$ is a vector and k is any scalar, then k $\vec{a}$ is defined to be a vector whose magnitude is |k| |$\vec{a}$ | whose direction is the same or opposite to $\vec{a}$ according as k is positive or negative. k $\vec{a}$ is called the scalar multiple of $\vec{a}$.

Thus if $\vec{a}$ is a vector, then -3 $\vec{a}$ is a vector whose magnitude is |-3| |$\vec{a}$| = 3 |$\vec{a}$| and whose direction is opposite to that of $\vec{a}$.

Scalar multiplication satisfies the following properties:
If m and n are any scalars and $\vec{a}$ and $\vec{b}$ are vectors then
  1. (m + n ) $\vec{a}$ = m$\vec{a}$ + n$\vec{a}$
  2. m($\vec{a}$ + $\vec{b}$) = m$\vec{a}$ + m$\vec{b}$
  3. m(n$\vec{a}$ ) = n(m$\vec{a}$ ) = mn $\vec{a}$
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Scalar Cross Product

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Cross product of two vectors is the area of the parallelogram between them and a cross product has both magnitude and direction.
Vector cross product is defined as
A * B = AB sin$\theta$ $\hat{n}$
The vector $\hat{n}$ is a unit vector perpendicular to the plane formed by the two vectors. Cross product is distributive.
Cross product of any vector with itself is zero. That is, A * A = 0

Scalar Triple Product

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Let $\vec{a}$, $\vec{b}$, $\vec{c}$ be three given vectors. Then $\vec{a}$ . ($\vec{b}$ * $\vec{c}$) is called a scalar triple product. It is the scalar product of the vectors a and ($\vec{b}$ * $\vec{c}$) and hence the result is a scalar. With $\vec{a}$, $\vec{b}$, $\vec{c}$ we can also from scalar triple product such as $\vec{b}$ . ($\vec{c}$ * $\vec{a}$), $\vec{c}$ . ($\vec{a}$ * $\vec{b}$)

Scalar triple product can be evaluated in two steps by first calculating b * c followed by a dot product with a or as below.

a. b * c = $\begin{vmatrix}
a_{1} & a_{2} &a_{3} \\
b_{1} &b_{2} &b_{3} \\
c_{1}&c_{2} &c_{3}
\end{vmatrix}$

Scalar triple product has the following properties.
  1. $\vec{a}$ . ( $\vec{b}$ * $\vec{c}$) = $\vec{b}$ . $\vec{c}$ * $\vec{a}$ = $\vec{c}$ . ($\vec{a}$ * $\vec{b}$)
  2. Scalar product is zero if any two vectors are identical.
  3. If $\vec{a}$, $\vec{b}$, $\vec{c}$ are coplanar then $\vec{a}$ . $\vec{b}$ * $\vec{c}$ = 0 and conversely.

Solved Example

Question: Find the scalar triple product $\vec{a}$ . $\vec{b}$ * $\vec{c}$ given that $\vec{a}$ = $\vec{i}$ - $\vec{j}$ + 2 $\vec{k}$, $\vec{b}$ = 2 $\vec{i}$ + 3 $\vec{k}$ and $\vec{c}$ = 2 $\vec{i}$ - 4 $\vec{j}$ + $\vec{k}$
Solution:
 
Given $\vec{a}$ = $\vec{i}$ - $\vec{j}$ + 2 $\vec{k}$, $\vec{b}$ = 2 $\vec{i}$ +3 $\vec{k}$ and $\vec{c}$ = 2 $\vec{i}$ - 4 $\vec{j}$ + $\vec{k}$

$\vec{a}.\vec{b}*\vec{c} = \begin{vmatrix}
1 &  -1& 2\\
 2& 0 &3 \\
2 & -4 & 1
\end{vmatrix}$

= 1 (12) - 1 (4) + 2 (-8)
= 12 - 4 - 16
= - 8
 

Scalar Dot Product

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If $\vec{a}$ and $\vec{b}$ are two vectors then their scalar product or dot product denoted by $\vec{a}$.$\vec{b}$ is defined by
$\vec{a}$.$\vec{b}$ = |$\vec{a}$|.|$\vec{b}$| cos $\theta$, where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$.

The name scalar product is due to the fact that $\vec{a}$.$\vec{b}$ is a scalar and the name dot product is due to the notation of placing a dot between the two vectors while denoting the scalar product.

The scalar product of two vectors satisfies the following properties.
  1. $\vec{a}$ . $\vec{b}$ = $\vec{b}$ . $\vec{a}$
  2. If $\vec{a}$ and $\vec{b}$ are both non zero vectors then $\vec{a}$ . $\vec{b}$ = 0 $\Rightarrow$ $\vec{a}$ and $\vec{b}$ are at right angles.
  3. If $\vec{a}$ and $\vec{b}$ are collinear vectors, then $\vec{a}$ . $\vec{b}$ = $\pm$ |$\vec{a}$| |$\vec{b}$|
  4. (m$\vec{a}$) . $\vec{b}$ = m ($\vec{a}$ . $\vec{b}$), if m is a scalar
  5. If $\vec{a}$ is any vector then $\vec{a}$ . $\vec{a}$ = |$\vec{a}$|$^{2}$
  6. If $\vec{a}$ and $\vec{b}$ are any two vectors and $\theta$ is the angle between them, then cos $\theta$ = $\frac{\vec{a}.\vec{b}}{|\vec{a}|.|\vec{b}|}$
  7. Dot product is distributive over vector addition.

Scalar Addition

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We can add two vectors after breaking them into x and y parts. Lets see with the help of examples how to find the scalar addition of the vectors.

Solved Examples

Question 1: Add the vectors a = (9, 4) and b = (15, 8)
Solution:
 
Given : a = (9, 4) and b = (15, 8)
let c = a + b
$\rightarrow$ c = (9, 4) + (15, 8)
c = (9 + 15, 4 + 8)
c = (24, 12)

 

Question 2: Find $\vec{a}$ if 3 $\vec{a}$ + (- 2, 1, 5) = 2 $\vec{a}$ - (1, 0, 1)
Solution:
 
3 $\vec{a}$ + (- 2$\vec{i}$, 1$\vec{j}$, 5$\vec{k}$) = 2 $\vec{a}$ - (1$\vec{i}$, 0$\vec{j}$, 1$\vec{k}$)
3 $\vec{a}$ - 2 $\vec{a}$ = - ( - 2$\vec{i}$, 1$\vec{j}$, 5$\vec{k}$) - (1$\vec{a}$, 0$\vec{j}$, 1$\vec{k}$)
$\vec{a}$ = $\vec{i}$ - $\vec{j}$ - 6$\vec{k}$