91Ó°ÊÓ

Vector addition is like combining two arrows' effects. If you have vectors \( \mathbf{u} \) and \( \mathbf{v} \), adding them means simply putting one after the other in a head-to-tail method. This way, it's similar to walking in one direction for \( \mathbf{u} \) units and then continuing for \( \mathbf{v} \) units.

Mathematically, you add their corresponding components. If \( \mathbf{u} = a\mathbf{i} + b\mathbf{j} \) and \( \mathbf{v} = c\mathbf{i} + d\mathbf{j} \), then:

• \( \mathbf{u} + \mathbf{v} = (a+c)\mathbf{i} + (b+d)\mathbf{j} \)

This combines the horizontal (\( \mathbf{i} \)) and vertical (\( \mathbf{j} \)) parts. For example, if \( \mathbf{u} = -1.1\mathbf{i} + 4\mathbf{j} \) and \( \mathbf{v} = 4\mathbf{i} + 2.4\mathbf{j} \), to find \( \mathbf{u} + \mathbf{v} \), we do:

• Add \(-1.1 + 4 = 2.9\) for the \( \mathbf{i} \) component.
• Add \(4 + 2.4 = 6.4\) for the \( \mathbf{j} \) component.

So, \( \mathbf{u} + \mathbf{v} = 2.9\mathbf{i} + 6.4\mathbf{j} \).

In vector subtraction, you're finding the difference or the gap between two vectors. Consider it like reversing the direction of one vector before adding. When you subtract vector \( \mathbf{v} \) from \( \mathbf{u} \), it calculates \( \mathbf{u} - \mathbf{v} \), which effectively involves flipping all components of \( \mathbf{v} \) and then adding.

If \( \mathbf{u} = a\mathbf{i} + b\mathbf{j} \) and \( \mathbf{v} = c\mathbf{i} + d\mathbf{j} \), then:

• \( \mathbf{u} - \mathbf{v} = (a-c)\mathbf{i} + (b-d)\mathbf{j} \)

This process ensures each respective component is solely influenced by the other vector's component. Using \( \mathbf{u} = -1.1\mathbf{i} + 4\mathbf{j} \) and \( \mathbf{v} = 4\mathbf{i} + 2.4\mathbf{j} \), we compute \( \mathbf{u} - \mathbf{v} \) by subtracting:

• Subtract \(-1.1 - 4 = -5.1\) from the \( \mathbf{i} \) component.
• Subtract \(4 - 2.4 = 1.6\) from the \( \mathbf{j} \) component.

Thus, \( \mathbf{u} - \mathbf{v} = -5.1\mathbf{i} + 1.6\mathbf{j} \).

Scalar multiplication involves stretching or shrinking a vector by a number, known as a scalar. This operation changes the magnitude (or length) of the vector, but not its direction, unless the scalar is negative, which also reverses the direction.

Let's take a vector \( \mathbf{u} = a\mathbf{i} + b\mathbf{j} \). If you multiply it by a scalar \( k \), then:

• \( k\mathbf{u} = (ka)\mathbf{i} + (kb)\mathbf{j} \)

This operation scales both components by \( k \). Considering \( \mathbf{u} = -1.1\mathbf{i} + 4\mathbf{j} \) and scaling by \( -3 \), we perform:

• -1.1 becomes \(-3 \times -1.1 = 3.3\).
• 4 becomes \(-3 \times 4 = -12\).

So, the resulting vector is \( -3\mathbf{u} = 3.3\mathbf{i} - 12\mathbf{j} \). This vector points in the opposite direction and is stretched by 3 times the magnitude of \( \mathbf{u} \). In operations like \(-3\mathbf{u} + \mathbf{v} \), you would then add \( \mathbf{v} = 4\mathbf{i} + 2.4\mathbf{j} \) to this scaled vector.