Vector Operations

Vectors are quantities that have both magnitude and direction. While there are various operations that can be applied to vectors, performing mathematical operations on them directly is not always possible. Therefore, special operations are defined specifically for vector quantities, known as vector operations.

It is crucial to understand the types of operations that can be performed on vector quantities, and vector operations provide insight into these possibilities.

Key Vector Operations:

1) Addition of Two Vectors
2) Subtraction of Two Vectors
3) Multiplication of a Vector by a Scalar
4) Product of Two Vectors : [Dot Product] & [Cross-Product]

Let's take a closer look at the vector operations mentioned above, along with the solved examples and illustrations in the article below.

Addition of Vectors

Vectors cannot be added by the usual algebraic rules. While adding two vectors, the magnitude and the direction of the vectors must be taken into account. The triangle law is used to add two vectors. The diagram below shows two vectors "a" and "b," and the resultant is calculated after their addition. Vector addition follows the commutative property, which means that the resultant vector is independent of the order in which the two vectors are added. 

\vec{a} + \vec{b} = \vec{c}

The commutative property of vector addition states that,

\vec{a} + \vec{b} = \vec{b} + \vec{a}

Triangle Law of Vector Addition

For the Triangle Law of Vector Addition, consider the vectors given in the figure above. The line AB represents the vector "a", and BC represents the vector "b". The line AC represents the resultant vector. The direction of AC is from A to C.

Line AC represents, 

\vec{a} + \vec{b}

The magnitude of the resultant vector is given by, 

\sqrt{|a|^2 + |b|^2 + 2|a||b|cos(\theta)}

The θ represents the angle between the two vectors. Let Φ be the angle made by the resultant vector with the vector p.

tan(\phi) = \frac{qsin(\theta)}{p + qcos(\theta)}

Parallelogram Law of Vector Addition

According to the Parallelogram Law of "The Adjacent sides of a parallelogram represent two vectors, then the diagonal starting from the same initial point represents the resultant of the vectors."

This is represented by the image added below:

Here, vector A and vector B represent the sides of parallelogram PQ and QR, respectively, and QS represents the resultant sum vector.

Subtraction of Two Vectors

Two vectors can be easily subtracted using the vector addition rules. A negative vector is considered a vector with an opposite direction, so it is easily solved by reversing its direction and applying the Triangle Law of Vector Addition.

Two vectors a and b are subtracted easily as shown in the image added below:

Multiplication of Vectors by a Scalar

Multiplying a vector a by a constant scalar k gives a vector whose direction is the same, but the magnitude is changed by a factor of k.

The figure shows the vector before and after it is multiplied by the constant k. In mathematical terms, this can be rewritten as, 

|k\vec{a}|~=~k|\vec{a}| 

if k > 1, the magnitude of the vector increases while it decreases when k < 1. The image added below shows the scalar multiplication of vector a with a scalar number k, where k is any constant greater than 1. (k>1)

Product of Two Vectors

Vectors can be multiplied by each other, but they cannot be divided. In the case of multiplication, there are two kinds of multiplication,

• Scalar Multiplication of Vector Or Dot Product
• Vector Multiplication of a Vector

Scalar Multiplication (also known as the dot product) is a kind of multiplication that results in a scalar quantity. Vector Multiplication (also known as Cross Product) is a kind of multiplication that results in a vector quantity.

Dot Product Or Scalar Product of Vector

Consider two \vec{A}  and \vec{B} . The scalar product of these two vectors is defined by the equation, 

\vec{A}\cdot \vec{B} = |\vec{A}||\vec{B}|cos(\theta)

Here, θ is the angle between two vectors.

In case the vectors are given by their components. for example a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k. In this case, the dot product is given by, 

a.b = a₁b₁i + a₂b₂j + a₃b₃k

Vector Product Or Cross Product of Vectors

Consider two vectors \vec{A}  and \vec{B}. The vector product of these two vectors is denoted by \vec{A} \times \vec{B}. The direction of this vector is perpendicular to both of the vectors. The magnitude of this vector is given by, 

|\vec{A} \times \vec{B}| = |\vec{A}||\vec{B}|sin(\theta)

Here, θ is the angle between two vectors. 

The right-hand rule is used to determine the direction of the resulting vector from the cross-product. Note that, unlike the addition and dot product, the vector product is not commutative.

In case the vectors are given by their components. for example a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k. In this case, the cross-product is given by,

\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} \\ 

Read More,

• Scalars and Vectors
• Product of Vectors
• Properties of Vectors
• Difference between vectors and scalar quantities

Solved Examples of Vector Operations

Problem 1: A vector is given magnitudes, v = 2i + j. Find the magnitude of the vector when it is scaled by a constant of 0.4. 

Solution: 

For any vector (v) = ai + bj, its magnitude is given as

|v| = \sqrt{a^2 + b^2}the

0.4|v| = |0.4v| 

a = 2, b = 1

|0.4v| = |0.4(2i + j)| 

⇒ |0.4v| = |0.8i + 0.4j| 

⇒ |0.4v| = \sqrt{0.8^2 + 0.4^2}

⇒ |0.4v| = \sqrt{0.64 + 0.16}

⇒ |0.4v| = 0.8

Hence, the magnitude of the vector, v = 2i + j when it is scaled by a constant of 0.4 is 0.8

Problem 2: Two vectors with magnitudes 5 and 10. These vectors have a 60° angle between them. Find the magnitude of the resultant vectors. 

Solution: 

Let, two vectors be p and q. Then resultant vector "r" is given by, 

|r| = \sqrt{|p|^2 + |q|^2 + 2|p||q|cos(\theta)}

Given,

• |p| = 5
• |q| = 10
• θ = 60^o

⇒ |r| = \sqrt{|p|^2 + |q|^2 + 2|p||q|cos(\theta)}

⇒ |r| = \sqrt{|5|^2 + |10|^2 + 2|5||10|cos(60)}

⇒ |r| = \sqrt{|5|^2 + |10|^2 + (10)(5)}

⇒ |r| = \sqrt{25 + 100 + 50}

⇒ |r| = \sqrt{175}

The magnitude of resultant vector is |r| = \sqrt{175}

Problem 3: Two vectors with magnitudes 4 and 4. These vectors have a 60° angle between them. Find the magnitude of the resultant vectors and the angle made by the resultant vector. 

Solution: 

Let, two vectors be p and q. Then resultant vector "r" is given by, 

|r| = \sqrt{|p|^2 + |q|^2 + 2|p||q|cos(\theta)}

• |p| = 4
• |q| = 4
• θ = 60^o

|r| = \sqrt{|p|^2 + |q|^2 + 2|p||q|cos(\theta)}

⇒ |r| = \sqrt{|4|^2 + |4|^2 + 2|4||4|cos(60)}

⇒ |r| = \sqrt{16 + 16 + 2|4||4|cos(60)}

⇒ |r| = \sqrt{48}

Angle made by resultant, 

tan(\phi) = \frac{4sin(60)}{4 + 4cos(60)}

⇒ tan(\phi) = \frac{2\sqrt{2}}{4 + 2}

⇒ tan(\phi) = \frac{\sqrt{2}}{3}

⇒ Φ = tan^-1(2/3)

Problem 4: Two vectors are given by a = 2i + j + k and b = i + j + k. Find the dot product of these two vectors. 

Solution: 

Given,

• a = 2i + j + k 
• b = i + j + k 

Dot Product = a.b 

⇒ a.b = (2i + j + k ).(i + j + k)

⇒ a.b = 2.1 i² + 1.1 j²+ 1.1 k²

⇒ a.b = 2 + 1 + 1 = 4 

Hence, dot product of the given two vectors is 4.

Problem 5: Two vectors are given as aaby2i + j + k and b = i + j + k. Find the cross product of these two vectors. 

Solution: 

Given,

• a = 2i + j + k 
• b = i + j + k 

Cross Prodcut of Vector = (2i + j + k) × (i + j + k)

\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 1 & 1 \\ 1 & 1 & 1 \end{vmatrix}

\Rightarrow \vec{A} \times \vec{B} = \hat{i}\begin{vmatrix} 1 & 1 \\ 1 & 1 \end{vmatrix} - \hat{j}\begin{vmatrix} 2 & 1 \\ 1 & 1 \end{vmatrix} + \hat{k}\begin{vmatrix} 2 & 1 \\ 1 & 1 \end{vmatrix}

\Rightarrow \vec{A} \times \vec{B} = -\hat{j}(2.1 - 1.1) + \hat{k}(2.1 - 1.1)

\Rightarrow \vec{A} \times \vec{B} = -\hat{j} + \hat{k}

Practice Problem Based on Vector Operations

Question 1. Two vectors are given by \vec{a} = 3\hat{i} + 2\hat{j} + 4\hat{k} and \vec{b} = \hat{i} + 3\hat{j} + \hat{k} . Find the dot product of these two vectors .

Question 2. Given two vectors\vec{a} = 4\hat{i} - \hat{j} + 2\hat{k} and \vec{b} = 2\hat{i} + \hat{j} + 3\hat{k}, find the magnitude of the cross product.

Question 3. Two vectors \vec{a} = \hat{i} + 2\hat{j} and \vec{b} = 2\hat{i} + 3\hat{j} are given. Find the angle between the two vectors.

Question 4. Find the magnitude of the resultant vector where, \vec{a} = 3\hat{i} - \hat{j} and \vec{b} = 2\hat{i} + 4\hat{j} .

Answer:-

1. 13
2. \sqrt{101}
3. \theta = \cos^{-1}\left(\frac{8}{\sqrt{65}}\right) \approx 50.24^\circ
4. |\vec{r}| = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34} \approx 5.83