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Resultant of Perpendicular Vectors

Resultant of Perpendicular Vectors

In previous lessons, you learnt about the resultant vector in one dimension, we are going to extend this to two dimensions. As a reminder, if you have a number of vectors (think forces for now) acting at the same time you can represent the result of all of them together with a single vector known as the resultant. The resultant vector will have the same effect as all the vectors adding together.

We will focus on examples involving forces but it is very important to remember that this applies to all physical quantities that can be described by vectors, forces, displacements, accelerations, velocities and more.

Vectors on the Cartesian Plane

The first thing to make a note of is that in Grade 10 we worked with vectors all acting in a line, on a single axis. We are now going to go further and start to deal with two dimensions. We can represent this by using the Cartesian plane which consists of two perpendicular (at a right angle) axes. The axes are a \(x\)-axis and a \(y\)-axis. We normally draw the \(x\)-axis from left to right (horizontally) and the \(y\)-axis up and down (vertically).

We can draw vectors on the Cartesian plane. For example, if we have a force, \(\vec{F}\), of magnitude \(\text{2}\) \(\text{N}\) acting in the positive \(x\)-direction we can draw it as a vector on the Cartesian plane.

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Notice that the length of the vector as measured using the axes is \(\text{2}\), the magnitude specified. A vector doesn’t have to start at the origin but can be placed anywhere on the Cartesian plane. Where a vector starts on the plane doesn’t affect the physical quantity as long as the magnitude and direction remain the same. That means that all of the vectors in the diagram below can represent the same force. This property is know as equality of vectors.

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In the diagram the vectors have the same magnitude because the arrows are the same length and they have the same direction. They are all parallel to the \(x\)-direction and parallel to each other.

This applies equally in the \(y\)-direction. For example, if we have a force, \(\vec{F}\), of magnitude \(\text{2.5}\) \(\text{N}\) acting in the positive \(y\)-direction we can draw it as a vector on the Cartesian plane.

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Just as in the case of the \(x\)-direction, a vector doesn’t have to start at the origin but can be placed anywhere on the Cartesian plane. All of the vectors in the diagram below can represent the same force.

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The following diagram shows an example of four force vectors, two vectors that are parallel to each other and the \(y\)-axis as well as two that are parallel to each other and the \(x\)-axis.

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To emphasise that the vectors are perpendicular you can see in the figure below that when originating from the same point the vector are at right angles.

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Vectors in two dimensions are not always parallel to an axis. We might know that a force acts at an angle to an axis so we still know the direction of the force and if we know the magnitude we can draw the force vector. For example, we can draw \(\vec{F}_{1} = \text{2}\text{ N}\) acting at \(\text{45}\)\(\text{°}\) to the positive \(x\)-direction:

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We always specify the angle as being anti-clockwise from the positive \(x\)-axis. So if we specified an negative angle we would measure it clockwise from the \(x\)-axis. For example, \(\vec{F}_{1} = \text{2}\text{ N}\) acting at \(-\text{45}\)\(\text{°}\) to the positive \(x\)-direction:

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We can use many other ways of specifying the direction of a vector. The direction just needs to be unambiguous. We have used the Cartesian coordinate system and an angle with the \(x\)-axis so far but there are other common ways of specifying direction that you need to be aware of and comfortable to handle.

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