Representation of Vector in Coordinates and Graph

Learn how a vector is represented, see how its’magnitude and direction are shown, then explore its representation using coordinates and graphs — step by step.

Representation of Vector in Coordinates and Graph

A vector can be represented in different forms to clearly show its magnitude and direction. In mathematics, two common ways of representing a vector are through coordinates and a graph.

Coordinates – represent a vector using an ordered pair of numbers that show its horizontal and vertical components.

Graph – represents a vector visually on a coordinate plane using a directed line segment or arrow, where the length shows its magnitude and the arrowhead shows its direction.

These representations help us understand the movement, magnitude, and direction of a vector on a coordinate plane.

Representation of Vector in a Coordinates

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Generally, vectors are denoted by bold face letters in writing. They are shown by placing an arrow over the letter.

In the figure, OA can be denoted by a or a. a is read as “vector a”.

In the diagram, the horizontal displacement of OA is x and the vertical displacement is y.

Therefore, in coordinates, vector OA = (x, y) in row and (xy) in column form.

Here, x is called the x-component and y is called the y-component.

In the figure, when points P(x₁, y₁) and Q(x₂, y₂) are joined, the vector PQ is formed.

From points P and Q, perpendiculars PM and QN are drawn to the X-axis, and from P, PR ⊥ QN is drawn.

Then, PR = MN = x₂ – x₁, QR = y₂ – y₁.

Here, the horizontal displacement (x-component) of PQ (b) is PR, and the vertical displacement (y-component) is QR.

Therefore, in coordinates, PQ is written as (x₂ – x₁, y₂ – y₁) in row and (x₂ – x₁y₂ – y₁) in column form.

Here, x-component of PQ is x₂ – x₁, y-component of PQ is y₂ – y₁.

Representation of Vector in a Graph

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Plot the following points on the graph:

O(0, 0), A(2, 3), B(–3, 2), C(4, 2) and join the line segments OA, OC and AB.

In the adjoining figure, the line segments OA, OC and AB are shown.

If a displacement occurs from point O(0, 0) to A(2, 3), this displacement is represented by the directed line segment OA.

Here, O is the initial point and A is the terminal point after displacement.

Therefore, OA is a vector.

OA represents displacement from O to A, whereas AO represents displacement from A to O.

Thus, OA and AO are different vectors.

Similarly, OC and AB are vectors.

In the figure, if a point moves from O(0, 0) to A(x, y), the displacement is represented by the directed segment OA.

Here, O is the initial point and A is the terminal point after displacement.

Hence, OA is a vector.

OA shows the displacement from O to A.

Example 1 of 2

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