Equivalent Vectors

Subsection 2.8.3 Equivalent Vectors

Now that you understand how to represent vectors using magnitude and direction, we can explore how to compare them. Is it possible for two vectors to be considered “equal” even if they are located in different places on a graph? In this section, you will examine the specific conditions regarding length and direction that must be met for two vectors to be called equivalent.

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Activity 2.8.3.

Work in groups

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What you require: Graph paper,ruler

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(a)

Draw the \(x\) and \(y\) axis on the graph paper.

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(b)

Plot the points \(A(0,4), B(3,4), C(0,2)\) and \(D(3,2)\text{.}\)

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(c)

Draw a line to connect point \(A\) and \(B,\) add an arrow pointing to point \(B.\)

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(d)

Draw a line to connect point \(C\) and \(D,\) add an arrow pointing to point \(D.\)

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(e)

Look at the two arrows you have drawn. Do they look like "clones" (exact copies) of each other?

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(f)

Imagine sliding the vector \(\mathbf{AB}\) straight down without turning it.

If you move point \(A\) so it sits exactly on top of point \(C\text{,}\) where does point \(B\) land?
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Does it land exactly on point \(D\text{?}\)
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(g)

Calculate the length of both vectors and compare the direction they are pointing. What two properties do vector \(\mathbf{AB}\) and vector \(\mathbf{CD}\) have in common?

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(h)

Since these vectors share the exact same properties, discuss with your group what relationship exists between them.

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Key Takeaway 2.8.8.

Two or more vectors are said to be equivalent if they satisfy the following conditions:

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They have same magnitude.
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They point in the same direction.
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Figure 2.8.9.

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In Figure 2.8.9 above, \(\textbf{MN} = \textbf{PQ}\) since they have same direction and equal magnitude.

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Example 2.8.10.

Using Figure 2.8.11 below, determine whether vector \(\mathbf{AB}\) and \(\mathbf{DC}\) are equivalent.

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Figure 2.8.11.

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Solution.

Vector \(\mathbf{AB} \text{ and } \mathbf{DC}\) are equivalent because they have the same magnitude, \(|\mathbf{AB}| = |\mathbf{DC}| \text{,}\) and they point in the same direction

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