Angle Between Two Vectors

The angle between two vectors Vector and Vector is given by the formula:

Angle Between Two Vectors Formula


Vectors

Angle Between Two Vectors

Angle


Examples

Calculate the dot product and the angle formed by the following vectors:

1. Vector = (3, 4) and Vector = (−8, 6)

Vector · Vector = 3 · (−8) + 4 · 6 = 0

Angle Between Two Vectors

2. Vector = (5, 6) and Vector = (−1, 4)

Vector · Vector = 5 · (−1) + 6 · 4 = 19

Angle Between Two Vectors

3. Vector = (3, 5) and Vector = (−1, 6)

Vector · Vector = 3 · (−1) + 5 · 6 = 27

Angle Between Two Vectors


Given the vectors Vector = (2, k) and Vector = (3, −2), calculate the value of k so that the vectors Vector and Vector are:

1 Perpendicular.

Vector Calculations

Vector Solution


2 Parallel.

Vector Calculations

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Vector Solution


3 Make an angle of 60°.

Vector Calculations

Vector Solution


Find the value of k if the angle between Vector = (3, k) and Vector = (2, −1) is:

1 90°

Vector Calculations

Vector Solution


2

Vector Calculations

Vector Solution


3 45°

Vector Calculations

Vector Calculations

Vector Solution



Calculate the angles of the triangle with vertices: A = (6,0), B = (3,5) and C = (−1,−1).

Angles of the Triangle

Vector Calculations

Vector Calculations

Vector Calculations

Vector Calculations

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Vector Calculations

Vector Calculations

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Vector Calculations

Vector Solution