## Video Lesson: How to Calculate the Magnitude and Direction of a Vector

## How to Calculate the Magnitude of a Vector

**The magnitude of a vector is a measure of its size or length. To calculate the magnitude, use the formula |v| = √(x ^{2} + y^{2}) where x and y are the horizontal and vertical components of the vector respectively.**

Formula for calculating the magnitude of a vector

The magnitude of a vector is equal to its length and is a scalar quantity. The magnitude does not describe the direction of the vector.

The magnitude of a vector cannot be negative. All vector magnitudes are positive except for the zero vector which has a magnitude of zero.

### How to Calculate the Magnitude of a 2D Vector

**The magnitude of a 2D vector is calculated with |v| = √(x^{2} + y^{2}), where x and y are the components of the vector.**

**For example, the magnitude of the vector v=(1, 4) is |v| = √(1^{2} + 4^{2}) = √17.**

The square root of 17 is approximately 4.12 and so, the length of the vector (1, 4) is approximately 4.12.

It is common to leave the magnitude of a vector written as a surd inside the square root in exact form.

Here are some examples of how to calculate the magnitude of a 2 dimensional vector:

Vector | Magnitude Calculation | Magnitude |

(5, 1) | √(5^{2}+1^{2}) | √26 |

(3, 4) | √(3^{2}+4^{2}) | 5 |

(0, 0) | √(0^{2}+0^{2}) | 0 |

(-1, -2) | √((-1)^{2}+(-2)^{2}) | √5 |

(2, 2) | √(2^{2}+2^{2}) | √8 |

### How to Calculate the Magnitude of a 3D Vector

**To calculate the magnitude of a 3D vector v=(x, y, z), use the formula |v| = √(x ^{2}+y^{2}+z^{2}).**

**For example, the magnitude of the 3D vector (2, -1, 3) is √(2^{2}+(-1)^{2}+3^{2}) which equals √14.**

The magnitude of a 3 dimensional vector is calculated using the Pythagorean theorem in the same way as a 2D vector but it involves one extra z component.

- x is the x-component of the vector
- y is the y-component of the vector
- z is the z-component of the vector

Formula for the magnitude of a 3 dimensional vector

Here are some examples of calculating the magnitude of a 3D vector:

Vector | Magnitude Calculation | Magnitude |

(1, 3, 5) | √(1^{2}+3^{2}+5^{2}) | √35 |

(3, 0, 1) | √(3^{2}+0^{2}+1^{2}) | √10 |

(-1, -2, -3) | √((-1)^{2}+(-2)^{2}+(-3)^{2}) | √14 |

(0,0,0) | √(0^{2}+0^{2}+0^{2}) | 0 |

(10, 10, 10) | √(10^{2}+10^{2}+10^{2}) | √300 |

## How to Calculate the Direction of a Vector

**To calculate the direction of the vector v = (x, y), use the formula θ = arctan(y/x).**

**For vectors in quadrant 1, the direction is θ, in quadrant 2 the direction is 180° – θ, in quadrant 3 the direction is 180° + θ and in quadrant 4 the direction is 360° – θ**.

The direction of a vector is the angle measured counter-clockwise from the positive x-axis direction. The angle **θ** is the angle made between the x and y components of the vector.

Therefore the direction of the vector depends on the quadrant that the vector is in.

### How to Calculate the Direction of a 2D Vector

Formula for the direction of a 2 dimension vector

For example, calculate the direction of the vector v = (1, 4).

- The x component is 1
- The y component is 4

Therefore and so, .

Since the vector is in the first quadrant, is the direction of the vector.

Calculate the direction of the vector (-2, -3).

- The x component of the vector is -2
- The y component of the vector is -3

The vector is in quadrant two and therefore, the direction is equal to .

and so, .

Evaluating this, and therefore the direction is .

## Vector Magnitude and Direction Calculator

The following calculator will calculate the magnitude and direction of a vector.

For any vector , enter the x component and y component in the calculator below to obtain the magnitiude and direction.