Operations on Vector

The operation by which two or more vectors or a scalar and a vector combine to give a single vector or a scalar is known as a vector operation. Here, in this level, we shall discuss the following operation,

1. Addition of Vector
2. Subtraction of vector
3. Multiplication of a vector by a scalar.

Addition of vectors

Let \(\overrightarrow{a}\) = \(\begin{pmatrix}x_1\\y_1\\\end{pmatrix}\) and\(\overrightarrow{b}\) =\(\begin{pmatrix}x_2\\y_2\\\end{pmatrix}\), then \(\overrightarrow{a}\) +\(\overrightarrow{b}\) = \(\begin{pmatrix}x_1\\y_1\\\end{pmatrix}\) +\(\begin{pmatrix}x_2\\y_2\\\end{pmatrix}\) =\(\begin{pmatrix}x_1+x_2\\y_1+y_2\\\end{pmatrix}\).

\(\therefore\) x-component of \(\overrightarrow{a}\) +\(\overrightarrow{b}\) = x₁ + x₂

and, y-component of \(\overrightarrow{a}\) +\(\overrightarrow{b}\) = y₁ + y₂

Hence, the addition of two column vectors is obtained by adding the corresponding x components and y components of \(\overrightarrow{a}\) and \(\overrightarrow{b}\).

Triangle Law of Vector Addition

Let \(\overrightarrow{AB}\) displace point A to point B \(\overrightarrow{BC}\) displace point B to point C. Then, the total displacement from A to C i.e. \(\overrightarrow{AC}\) is given by\(\overrightarrow{AC}\) +\(\overrightarrow{BC}\).

This law of addition of vector is known as triangle law of vector addition and\(\overrightarrow{AC}\) is called the resultant vector.

Hence, triangle law of vector addition states that if two sides of a triangle taken in order tp represent two vectors then theirresultant will be given by the third side of the triangle in opposite order.

Parallelogram Law of a vector

Let \(\overrightarrow{AB}\) = \(\overrightarrow{a}\) and \(\overrightarrow{AD}\) = \(\overrightarrow{b}\) be two co-initial (having the same point of start) vectors with the initial points as A. A parallelogram ABCD is completed such that AC = DC and AD = BC. Also, we have , AB//CD and AD//BC.

\(\therefore\) \(\overrightarrow{DC}\) = \(\overrightarrow{AB}\) = \(\overrightarrow{a}\) and

\(\overrightarrow{AD}\) = \(\overrightarrow{BC}\) = \(\overrightarrow{b}\)

Now, by triangle law of vector addition we have,

\(\overrightarrow{AB}\) + \(\overrightarrow{BC}\) = \(\overrightarrow{AC}\)

or, \(\overrightarrow{AB}\) + \(\overrightarrow{AD}\) = \(\overrightarrow{AC}\)(\(\therefore\) \(\overrightarrow{AD}\) = \(\overrightarrow{BC}\))

\(\therefore\) \(\overrightarrow{a}\) + \(\overrightarrow{b}\) = \(\overrightarrow{AC}\)

This law of addition of vector is called parallelogram law of vector addition. Hence, the parallelogram law of vector adition states that if two adjacent sides of a parallelogram represent two co-initial vectors then the diagonal of the parallelogram passing through the same point gives its resultant.

Polygon Law of Vector Addition

Polygon law of vector addition is the generalization of the triangle law of vector addition where \(\overrightarrow{AB}\) + \(\overrightarrow{BC}\) + \(\overrightarrow{CD}\) + \(\overrightarrow{DE}\) = \(\overrightarrow{AE}\).

Proof:

ABCDE is a polygon (pentagon). Let's join AC and AD.

Now using \(\triangle\) law of vector addition in \(\triangle\)ABC, \(\triangle\)ADC and \(\triangle\)ADE, we get

\(\overrightarrow{AB}\) + \(\overrightarrow{BC}\) = \(\overrightarrow{AC}\).......(1)

\(\overrightarrow{AC}\) + \(\overrightarrow{CD}\) = \(\overrightarrow{AD}\).........(2)

Using (1) and (2)

\(\overrightarrow{AB}\) + \(\overrightarrow{BC}\) + \(\overrightarrow{CD}\) = \(\overrightarrow{AD}\).......(3)

Again, \(\overrightarrow{AD}\) + \(\overrightarrow{DE}\) = \(\overrightarrow{AE}\) .....()

using (3) and (4)

\(\overrightarrow{AB}\) + \(\overrightarrow{BC}\) + \(\overrightarrow{CD}\) + \(\overrightarrow{AD}\) + \(\overrightarrow{DE}\) = \(\overrightarrow{AE}\) proved.

Hence, the polygon law of vector addition states that if a number of vectors are represented by the sides of a polygon taken in order then the resultant vector is represented by the closing side of the polygon in the opposite order.

Subtraction of vectors

Let \(\overrightarrow{a}\) = \(\begin{pmatrix}a_1\\a_2\\\end{pmatrix}\) and\(\overrightarrow{b}\) =\(\begin{pmatrix}b_1\\b_2\\\end{pmatrix}\), then \(\overrightarrow{a}\) - \(\overrightarrow{b}\) = \(\begin{pmatrix}a_1\\b_2\\\end{pmatrix}\) - \(\begin{pmatrix}b_1\\b_2\\\end{pmatrix}\) =\(\begin{pmatrix}a_1-a_2\\b_1-b_2\\\end{pmatrix}\).

Subtraction is a reverse of addition. So. if AB and BC are two vectors then their difference of \(\overrightarrow{AB}\) - \(\overrightarrow{BC}\) is the sum of the vectors \(\overrightarrow{AB}\) and the negative of\(\overrightarrow{BC}\) i.e. -\(\overrightarrow{BC}\)

So,

\(\overrightarrow{AB}\) - \(\overrightarrow{BC}\) = \(\overrightarrow{AB}\) + \(\overrightarrow{BC}\)

Here,

\(\overrightarrow{AB}\) + \(\overrightarrow{BC}\) =\(\overrightarrow{AC}\)

But \(\overrightarrow{BD}\) is negative of \(\overrightarrow{BC}\).

i.e \(\overrightarrow{BD}\) = -\(\overrightarrow{BC}\)

or, \(\overrightarrow{AB}\) - \(\overrightarrow{BC}\)

= \(\overrightarrow{AB}\) + (-\(\overrightarrow{BC}\))

= \(\overrightarrow{AB}\) + \(\overrightarrow{BD}\)

= \(\overrightarrow{AD}\)

Multiplication of a vector by a scalar

\(\overrightarrow{a}\) = \(\begin{pmatrix}a_1\\a_2\\\end{pmatrix}\) k \(\overrightarrow{a}\) = k\(\begin{pmatrix}a_1\\a_2\\\end{pmatrix}\) =\(\begin{pmatrix}ka_1\\ka_2\\\end{pmatrix}\) where k is a scalar.

If k is a positive scalar, \(\overrightarrow{a}\) and \(\overrightarrow{a}\) have same direction. If k is a negative scalar\(\overrightarrow{a}\) and k\(\overrightarrow{a}\) will have opposite direction.