Operation of Vectors

Learn what vectors are, understand how they are represented, then explore vector operations — one step at a time.

Operations of Vectors

The vector

A vector is a quantity that has both magnitude and direction. It is commonly represented by an arrow or by its components:

A = Ax ı + Ay ȷ

where $A_x$ and $A_y$ are the components of the vector.

The same vector can be represented in different ways depending on what information we want to show.

The important operations of vectors are:

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

Addition of Vectors

Step 1 of 5
X Y O a = (2, 3) b = (4, 1) a + b = (6, 4)

Solution Steps

Introduction:

Vector addition is the process of adding two or more vectors by adding their corresponding components. Thus, the x-components are added together and the y-components are added together.

Example:

If a = (2, 3) and b = (4, 1), find the value of a + b.

Similarly, if c = (–2, 5) and d = (3, –1), find the value of c + d.

Solution

a + b = (2, 3) + (4, 1)

= (2 + 4, 3 + 1)

= (6, 4)

Similarly,

c + d = (–2, 5) + (3, –1)

= (–2 + 3, 5 – 1)

= (1, 4)

Thus, when adding vectors, add the corresponding x-components together and the corresponding y-components together.

Subtraction of Vectors

Step 1 of 6
X Y O a b Step 2: a and b both start at O. –b Step 3: –b is as long as b but points the opposite way. –b Step 4: Put –b at the tip of a. a – b Step 5: a – b = a + (–b), from O to the new tip.

Solution Steps

Introduction:

Vector subtraction is performed by subtracting the corresponding components of the vectors. It can also be written as the addition of a vector and the negative of another vector.

Example:

For example, if a = (x1y1) and b = (x2y2) then

–b = (–x2–y2)

Now, a – b = a + (–b)

= (x1y1) + (–x2–y2)

= (x1 – x2y1 – y2)

So, when subtracting vectors, subtract x-component from x-component and y-component from y-component.

Multiplication of a Vector by a Scalar

Step 1 of 6
X Y O 2a a Step 4: a = (2, 1) is the blue arrow. 2a = (4, 2) has the same direction but is twice as long. –a Step 5: (–1)a = (–2, –1) is as long as a but points the opposite way.

Solution Steps

Introduction:

Multiplying a vector by a scalar means multiplying each component of the vector by that scalar. The new vector is parallel to the original vector. Its length changes by the size of the scalar, and its direction reverses if the scalar is negative.

Example:

For example, if a = (xy) and k is a scalar, then find ka.

ka = k (xy)

= (kxky)

Now, if a = (21) and k = 2, then

2a = (2 × 22 × 1) = (42)

If k = –1, then

(–1)a = (–1 × 2–1 × 1) = (–2–1)

So, when multiplying a vector by a scalar, multiply the x-component by the scalar and the y-component by the scalar.

Magnitude and Direction of a Vector

Step 1 of 9
X X' Y Y' O 3 2 a = (3, 2) 6 4 b = (6, 4)

Solution Steps

Magnitude:

For a = (3, 2):

| a | = √32 + 22 = √9 + 4 = √13 units

For b = (6, 4):

| b | = √62 + 42 = √36 + 16 = √52 units

Notice: | b | = 2 | a |, which matches our earlier observation.

[Since √52 = √4 × 13 = 2√13]

Direction:

The direction (angle with x-axis) of a vector v = (x, y) is:

θ = tan–1(yx)

For a:

θa = tan–1(23)

For b:

θb = tan–1(46) = tan–1(23)

So, a and b have the same direction.

Here, when the magnitude of a is multiplied by 2, we get b. Also, a and b have the same direction.

Or, b = (6, 4) = 2(3, 2) = 2a

Thus, multiplying vector a by a scalar (2) gives vector b.

If a = (x, y) is a vector and k is any scalar, then multiplying a by k gives: ka = k(x, y) = (kx, ky). Its magnitude is: | ka | = | k | · | a |. If k is positive, then a and ka have the same direction. If k is negative, then a and ka have opposite directions. Whether k is positive or negative, the vectors remain parallel. In the figure, vectors a and b have the same direction and b = 2a. Therefore, a and b are parallel vectors.

Example 1 of 2

Solve It