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Chapter 6: Problem 96

If two vectors are expressed in terms of \(\mathbf{i}\) and \(\mathbf{j}\), explain how to find their difference.

Short Answer

Expert verified
The difference between two vectors \(\mathbf{A} = a1\mathbf{i} + a2\mathbf{j}\) and \(\mathbf{B} = b1\mathbf{i} + b2\mathbf{j}\) is found by subtracting the corresponding components of \(\mathbf{B}\) from \(\mathbf{A}\) to produce a new vector \(\mathbf{C} = (a1 - b1)\mathbf{i} + (a2 - b2)\mathbf{j}\).

Step by step solution

01

Understand Vector Subtraction

The subtraction of vectors is similar to their addition. If we have two vectors \(\mathbf{A} = a1\mathbf{i} + a2\mathbf{j}\) and \(\mathbf{B} = b1\mathbf{i} + b2\mathbf{j}\), their difference or subtraction would be another vector \(\mathbf{C} = \mathbf{A} - \mathbf{B} = (a1 - b1)\mathbf{i} + (a2 - b2)\mathbf{j}\), where \(\mathbf{C}\) is the result vector.
02

Subtract Individual Components

In the above example, we performed subtraction component-wise: the \(\mathbf{i}\) components were subtracted from each other, and the \(\mathbf{j}\) components were subtracted from each other. The scalars (the numbers in front of the \(\mathbf{i}\) and \(\mathbf{j}\)) were subtracted, rather than the unit vectors themselves. Each component acts independently of the other.
03

Visualize Vector Subtraction

In vector subtraction, the direction of the resultant vector is important. Subtracting a vector is the same as adding its negative. This means to subtract a vector you add a vector of same magnitude but in the opposite direction. So, the difference between two vectors \(\mathbf{A}\) and \(\mathbf{B}\), can also be seen as a vector \(\mathbf{C}\) where \(\mathbf{C}\) = \(\mathbf{A} + (-\mathbf{B})\). This can be visualized graphically as well.

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