91Ó°ÊÓ

Vector addition is a fundamental operation where two vectors are combined to form a new vector, called the resultant vetor. Imagine vectors as arrows. The tail of the "arrows" can be placed on a common point, and to add the vectors, you join the head of the first vector to the tail of the second. In terms of mathematics, you add their corresponding components.
For example, if we have vectors \( \mathbf{u} = \mathbf{i} + \mathbf{j} \) and \( \mathbf{v} = 2\mathbf{i} - 3\mathbf{j} \), vector addition involves:

• Adding the \( \mathbf{i} \) components: \( \mathbf{i} + 2\mathbf{i} = 3\mathbf{i} \)
• Adding the \( \mathbf{j} \) components: \( \mathbf{j} - 3\mathbf{j} = -2\mathbf{j} \)

This results in the new vector: \( \mathbf{u} + \mathbf{v} = 3\mathbf{i} - 2\mathbf{j} \), indicating the cumulative effect of moving in the direction of each vector.

Vector subtraction is just as crucial as vector addition. It can also be thought of as adding a negative version of a vector to another vector. Essentially, you are finding out how one vector differs from another in terms of their direction and magnitude.
To subtract the vectors \( \mathbf{u} = \mathbf{i} + \mathbf{j} \) and \( \mathbf{v} = 2\mathbf{i} - 3\mathbf{j} \), follow the steps below:

• Subtract the \( \mathbf{i} \) components: \( \mathbf{i} - 2\mathbf{i} = -\mathbf{i} \)
• Subtract the \( \mathbf{j} \) components: \( \mathbf{j} - (-3\mathbf{j}) = \mathbf{j} + 3\mathbf{j} = 4\mathbf{j} \)

These calculations give the resultant vector as \( \mathbf{u} - \mathbf{v} = -\mathbf{i} + 4\mathbf{j} \). This expresses how the directional movement given by \( \mathbf{u} \) alters when "backtracking" by \( \mathbf{v} \).

Scalar multiplication involves scaling a vector by multiplying it with a scalar (a real number). This operation affects the vector's magnitude, stretching or shrinking it, while maintaining its direction if the scalar is positive.
In the case of vectors \( \mathbf{u} = \mathbf{i} + \mathbf{j} \) and \( \mathbf{v} = 2\mathbf{i} - 3\mathbf{j} \), performing scalar multiplication for the expression \( 2\mathbf{u} - 3\mathbf{v} \) involves:

• Multiplying \( \mathbf{u} \) by 2: results in \( 2\mathbf{i} + 2\mathbf{j} \)
• Multiplying \( \mathbf{v} \) by 3: results in \( 6\mathbf{i} - 9\mathbf{j} \)

Next, employ vector subtraction, \( 2\mathbf{u} - 3\mathbf{v} = (2\mathbf{i} + 2\mathbf{j}) - (6\mathbf{i} - 9\mathbf{j}) \) which simplifies to:
Subtract \( 6\mathbf{i} \) from \( 2\mathbf{i} \), and add \( 9\mathbf{j} \) to \( 2\mathbf{j} \):

• \( 2\mathbf{i} - 6\mathbf{i} = -4\mathbf{i} \)
• \( 2\mathbf{j} + 9\mathbf{j} = 11\mathbf{j} \)

The final resultant vector is \( -4\mathbf{i} + 11\mathbf{j} \), expressing the combined effect of scaling \( \mathbf{u} \) and \( \mathbf{v} \), and then performing subtraction.