91Ó°ÊÓ

Vector addition is a fundamental operation involving two vectors. When adding two vectors, you simply add their corresponding components. In our example, we have two vectors:

• \( \mathbf{a} = \langle -4, 5 \rangle \)
• \( \mathbf{b} = \langle 2, -8 \rangle \)

To perform vector addition:

• Add the first components: \( -4 + 2 = -2 \)
• Add the second components: \( 5 - 8 = -3 \)

Thus, the resultant vector after addition is \( \mathbf{a} + \mathbf{b} = \langle -2, -3 \rangle \). Vector addition is visually represented by placing the tail of the second vector at the head of the first vector, thereby forming a parallelogram.

Vector subtraction allows us to find the difference between two vectors by subtracting their corresponding components. Suppose we have vectors \( \mathbf{a} = \langle -4, 5 \rangle \) and \( \mathbf{b} = \langle 2, -8 \rangle \). When subtracting, we do the following:

• Subtract the first components: \( -4 - 2 = -6 \)
• Subtract the second components: \( 5 - (-8) = 13 \)

Thus, the result of the subtraction is \( \mathbf{a} - \mathbf{b} = \langle -6, 13 \rangle \). Graphically, vector subtraction is akin to adding the negative of a vector. You reverse the direction of \( \mathbf{b} \) and then add as in a regular addition operation.

Scalar multiplication involves multiplying a vector by a scalar, which scales the vector's magnitude without changing its direction, unless if the scalar is negative. Consider the vector \( \mathbf{a} = \langle -4, 5 \rangle \). To scale this vector by 2, perform the following:

• Multiply each component by 2: \( 2 \times -4 = -8 \) and \( 2 \times 5 = 10 \)

This gives us the vector \( 2 \mathbf{a} = \langle -8, 10 \rangle \). Now, let's apply a negative scalar to \( \mathbf{b} = \langle 2, -8 \rangle \), scaling by \(-\frac{1}{2}\):

• Each component is \(-\frac{1}{2} \times 2 = -1 \) and \(-\frac{1}{2} \times -8 = 4 \)

Thus, we have \(-\frac{1}{2} \mathbf{b} = \langle -1, 4 \rangle \). Notice the vector has been both scaled down and its direction changed due to the negative scalar.

Coordinate vectors simplify the representation and manipulation of vectors. They are especially useful in illustrating vector components within a Cartesian plane. For the vectors \( \mathbf{a} = \langle -4, 5 \rangle \) and \( \mathbf{b} = \langle 2, -8 \rangle \), the components \(-4\) and \(5\) for \( \mathbf{a} \) and \(2\) and \(-8\) for \( \mathbf{b} \) identify their positions in the plane. These coordinate vectors facilitate operations like addition, subtraction, and scalar multiplication through straightforward arithmetic.

• Each component corresponds to an axis: first for \(x\) and second for \(y\).

This systematic layout allows students to predict the outcome of operations by visualizing movement along the axes. It is also crucial in graphing vectors, merger of operations like addition, and geometrical interpretations.