# How to Find the Angle Between Two Vectors?

Physical quantities that have both directions and magnitude are vectors. A vector with magnitude equal to one and direction equal to one is a unit vector “û.”. That is the lower-case alphabet with a “hat” circumflex. In this way, vectors are described by arrows, have initial points and terminal points, and were developed over a period of 200 years. Vectors can be used to represent physical quantities, such as displacement, velocity, acceleration, etc.

### Angle Between Two Vectors

A vector’s angle between its tails is equal to its angle between two vectors. It can be obtained using a dot product (scalar product) or cross product (vector product). Note that the angle between the two vectors remains between 0° and 180°. The angle between vectors can be found by using two methods. But the most commonly used formula for finding an angle between two vectors involves the scalar product.

### Finding angle using Scalar (dot) product

Two vectors combined into a scalar product give you a number. Scalar products can be used to define the relationships between energy and work. In mathematics, a scalar product is used to represent the work done by a force (which is a vector) in dispersing (which is a vector) an object. The scalar product is represented by a dot (.). Let,

Dot product be (a.b)

Magnitude of vector a = |a|

Magnitude of vector b = |b|

**Angle between the vectors is θ = Cos ^{-1 }[(a · b) / (|a| |b|)]**

When two vectors are connected by a dot product, the direction of the angle ፀ does not matter. The angle ፀ can be measured by the difference between either vector since Cos ፀ = Cos (-ፀ) = Cos (2π – ፀ).

### Finding angle using cross (vector) product

A cross product may also be known as a vector product. It is a form of vector multiplication that takes place between two vectors that have different kinds or natures. When two vectors are multiplied with each other and the resulting product is also a vector quantity, the resulting vector is called the cross product of two vectors or the vector product. Multiplication of two vectors yields vector products with a direction perpendicular to each vector. Let,

Cross product be (a × b)

Magnitude of vector a = |a|

Magnitude of vector b = |b|.

|a × b| = |a| |b| sin θ

**Angle between the vectors is θ = Sin ^{-1} [|a × b| / (|a| |b|)]**

### Sample Problems

**Question 1: Find the angle between two vectors a = {4, 5} and b = {5, 4}.**

**Solution:**

- Finding dot product ( a.b) = 4 × 5 + 5 × 4 = 40.
- Finding vectors magnitude, |a| = = √41, |b| = = √41.
- Angle between vectors, θ = Cos
^{-1}[(a · b) / (|a| |b|)] , θ = Cos^{-1}[(40) / (√41 × √41)]Angle between a and b,

θ = Cos

^{-1}[(40) / (41)]

**Question 2: Find the angle between two vectors a = {2, 2} and b = {1, 1}.**

**Solution:**

- Finding dot product (a.b) = 2 × 1 + 2 × 1 = 4.
- Finding vectors magnitude, |a| = = √8, |b| = = √2.
- Angle between vectors, θ = Cos
^{-1}[(a · b) / (|a| |b|)], θ = Cos^{-1}[(4) / (√8 × √2)].Angle between a and b,

θ = Cos

^{-1}[(4) / (4)]θ = Cos

^{-1}[1] = 0°.This means both vectors are overlapping each other and are in same direction.

**Question 3: Find the angle between two vectors a = **i** + 2j – k and b = 2i + 4j – 2k.**

**Solution:**

- Finding dot product (a.b) = 1 × 2 + 2 × 4 + (-1) × (-2) = 2 + 8 + 2 = 12.
- Finding vectors magnitude, |a| = = √6, |b| = = √24.
- Angle between vectors, θ = Cos
^{-1}[(a · b) / (|a| |b|)] , θ = Cos^{-1}[(12) / (√6 × √24)].Angle between a and b,

θ = Cos

^{-1}[(12) / (12)]θ = Cos

^{-1}[1] = 0°.This means both vectors are overlapping each other and are in same direction.

**Question 4: Find the angle between two vectors a = i + 2j – k and b = 4j – 2k.**

**Solution:**

Vector b can be written as, b = 0i + 4j – 2k.

- Finding dot product (a.b) = 1 × 0 + 2 × 4 + (-1) × (-2) = 0 + 8 + 2 = 10.
- Finding vectors magnitude, |a| = = √6, |b| = = √20.
- Angle between vectors, θ = Cos
^{-1}[(a · b) / (|a| |b|)] , θ = Cos^{-1}[(10) / (√6 × √20)].Angle between a and b,

θ = Cos

^{-1}[(10) / (√120)].

**Question 5: Find the angle between two vectors a = {1, -3} and b = {-3, 1}.**

**Solution:**

- Finding cross product magnitude |a × b| = √(0)² + (0)² + (-8)² = 8.
- Finding vectors magnitude, | a| = = √10, |b| = = √10.
- Angle between vectors, θ = Sin
^{-1}[(|a × b|) / (|a| |b|)], θ = Sin^{-1}[(8) / (√10 × √10)]Angle between a and b,

θ = Sin

^{-1}[(8) / (10)]

**Question 6: Find the angle between two vectors a = -3i + j and b = -3i + j.**

**Solution:**

- Finding dot product (a.b) = (-3) × (-3) + 1 × 1 = 10.
- Finding vectors magnitude, |a| = = √10, |b| = = √10
- Angle between vectors, θ = Cos
^{-1}[(a · b) / (|a| |b|)], θ = Cos^{-1}[(10) / (√10 × √10)].Angle between a and b,

θ = Cos

^{-1}[(10) / (10)]θ = Cos

^{-1}[1] = 0°As both vectors are having same value and direction so they are same vectors hence having angle 0 between them.