 # Angle BetweenTwo Vectors in a 2Dsolver

Angle between two vectors calculator computes the angle between two vectors in a 2D space.
Follow steps below to solve the angle between two vectors in a 2D space.

### Step: Enter value of each vector

Enter the values of each vector assuming you are working with vectors in this form

### result

Hit the check mark to solve for vector in 2d

Angle between two vectors calculator computes the angle between two vectors in a 2D space. A 2D Vector is a vector geometry in 2-dimensions running from point A (tail) to point B (head). Each vector has a magnitude (or length) and direction and can be calculated by taking the square root of the sum of each components in space.

### Lesson Note on Angle Between Two Vectors

To illustrate the calculation of the angle between two vectors in a 2D space, let's consider the following example:

#### Example 1

Let's say we have two vectors in a 2D space:

• Vector A with magnitude 3 and direction 45 degrees counterclockwise from the positive x-axis.

• Vector B with magnitude 2 and direction 30 degrees counterclockwise from the positive x-axis.

Step 1: Represent the vectors graphically

Draw a coordinate system with the x-axis and y-axis. Place the tail of Vector A at the origin (0,0) and draw the arrow pointing towards point B. Similarly, place the tail of Vector B at the origin and draw the arrow pointing towards point C.

Step 2: Calculate the components of each vector

To calculate the components of Vector A, we need to determine its x-component (Ax) and y-component (Ay). Since Vector A is 45 degrees counterclockwise from the positive x-axis, we can use trigonometry to find these components:

Ax = magnitude of A cos(angle of A)
= 3
cos(45 degrees)
= 3 * sqrt(2) / 2
= 3sqrt(2) / 2

Ay = magnitude of A sin(angle of A)
= 3 sin(45 degrees)
= 3 * sqrt(2) / 2
= 3sqrt(2) / 2

Similarly, we can calculate the components of Vector B:

Bx = magnitude of B cos(angle of B)
= 2
cos(30 degrees)
= 2 * sqrt(3) / 2
= sqrt(3)

By = magnitude of B sin(angle of B)
= 2
sin(30 degrees)
= 2 * 1/2
= 1

Step 3: Calculate the dot product of the vectors

The dot product of two vectors is calculated by multiplying their corresponding components and summing them up:
Dot product (A · B) = Ax Bx + Ay By

= (3sqrt(2) / 2) sqrt(3) + (3sqrt(2) / 2) 1
= 3sqrt(6) / 2 + 3sqrt(2) / 2
= (3sqrt(6) + 3sqrt(2)) / 2

Step 4: Calculate the magnitudes of the vectors

The magnitude of a vector is the length or size of the vector and can be calculated using the Pythagorean theorem:
Magnitude of A = sqrt(Ax^2 + Ay^2)
= sqrt((3sqrt(2) / 2)^2 + (3sqrt(2) / 2)^2)
= sqrt(9/2 + 9/2)
= sqrt(9)
= 3

Magnitude of B = sqrt(Bx^2 + By^2)
= sqrt((sqrt(3))^2 + 1^2)
= sqrt(3 + 1)
= sqrt(4)
= 2

Step 5: Calculate the angle between the two vectors

The angle between two vectors can be found using the dot product and the magnitudes of the vectors:

Angle between A and B (θ) = cos^(-1)((A · B) / (magnitude of A * magnitude of B))

= cos^(-1)(((3sqrt(6) + 3sqrt(2)) / 2) / (3 * 2))
= cos^(-1)((3sqrt(6) + 3sqrt(2)) / 6)
= cos^(-1)(sqrt(6) + sqrt(2)) / 2

Therefore, the angle between Vector A and Vector B in this example is cos^(-1)((sqrt(6) + sqrt(2)) / 2).

#### Example 2: find the angle between two vectors in 2D space, you can use the dot product formula. The formula is as follows:

θ = arccos( (a · b) / (|a| * |b|) )

Step 1: Find the dot product of vectors a and b.

a · b = (3 2) + (4 8)

= 6 + 32

= 38

Step 2: Find the magnitude (length) of vector a.

|a| = √(3^2 + 4^2)

= √(9 + 16)

= √25

= 5

Step 3: Find the magnitude (length) of vector b.

|b| = √(2^2 + 8^2)

= √(4 + 64)

= √68

≈ 8.246

Step 4: Substitute the values in the formula to find the angle.

θ = arccos( 38 / (5 * 8.246) )

Step 5: Use a calculator to find the arccos of the number.

θ ≈ arccos(0.91892)

≈ 23.57 degrees (rounded to two decimal places)

Therefore, the angle between vectors a(3, 4) and b(2, 8) in 2D space is approximately 23.57 degrees.

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