How to Find the Direction of Vectors?

The direction of a vector is the angle it makes with the positive x-axis, found from its components as θ = arctan(y/x), with a quadrant adjustment, since arctan alone only returns angles between -90° and 90°. Below: the method, the quadrant rule, and worked examples.

A vector is a physical quantity that is described by magnitude and direction. A vector quantity is represented by a vector diagram and therefore has a direction, the orientation at which the vector points are denoted as the direction of a vector.

Related Topics

• How to Find Vector Components
• How to Find Magnitude of Vectors
• How to Find Scalar Multiplication of Vectors
• How to Find Addition and Subtraction of Vectors

A step-by-step guide to the direction of vectors

• A vector’s orientation, or the angle it creates with the \(x\)-axis, determines its direction. With an arrow at the top but another fixed point at the other end, a vector may be shown. The vector’s direction is determined by the arrowhead’s point of departure. Velocity is an example of a vector. It tells you how fast the object is traveling, as well as the direction it is moving in. It tells you how fast the object is traveling, as well as the direction it is moving in. The force vector also indicates the direction in which a force is being applied. Vectors are denoted by where the magnitude of the vector is denoted by and the direction of the vector is denoted by.

• The direction of travel may be determined using the following equation:

\(\color{blue}{θ=tan^{-1}\frac{y}{x}}\), where \(x\) is the horizontal change and \(y\) is the vertical change.

or

\(\color{blue}{θ=tan^{-1}\frac{y_2-y_1}{x_2-x_1}}\), where \((x_1,y_1)\) is the initial point and \((x_2,y_2)\) is the terminal point.

Note: When calculating the angle, the quadrant in which \((x, y)\) lies is also to be considered.

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The Direction of Vectors , Example 1:

Find the direction of the vector \((1, \sqrt{3})\).

Use this formula to find the direction of vectors: \(\color{blue}{θ=tan^{-1}\frac{y}{x}}\).

\( θ =tan^{-1}\frac{\sqrt{3}}{1}=\sqrt{3}\)

\( θ = tan^{-1}\sqrt{3}\)

\( θ=60^\circ\)

The Direction of Vectors , Example 2:

Find the direction of the vector which starts at \((1, 3)\) and ends at \((4, 6)\).

Use this formula to find the direction of vectors: \(\color{blue}{θ=tan^{-1}\frac{y_2-y_1}{x_2-x_1}}\).

\((x_1,y_1)=(1,3)\)

\((x_2,y_2)=(4,6)\)

\( θ =tan^{-1}\frac{6-3}{4-1}=\frac{3}{3}\)

\( θ =tan^{-1}\frac{3}{3}\)

\( θ =45^\circ\)

Exercises for Direction of Vectors

1. Find the direction of a vector whose coordinates are \((-4, 3)\).
2. Find the direction of a vector whose initial and final points are \((3, 2)\) and \((8, 5)\), respectively.
3. Find the direction of a vector whose initial and final points are \((2, 3)\) and \((-3, -2)\), respectively.
4. Find the direction of a vector whose coordinates are \((1, -\sqrt{3})\).

1. \(\color{blue}{θ=143.13^\circ}\)
2. \(\color{blue}{θ=30.96^\circ}\)
3. \(\color{blue}{θ=225^\circ}\)
4. \(\color{blue}{θ=300^\circ}\)