91Ó°ÊÓ

The magnitude of a vector, essentially its length, can be understood as the distance from the origin to the point represented by the vector in a coordinate system. This is applicable in both two and three dimensions, but we'll focus on two-dimensional vectors here for simplicity. To find the magnitude, you use the formula:

• If you have a vector \(\mathbf{v} = a\mathbf{i} + b\mathbf{j}\), then its magnitude \(|\mathbf{v}|\) is calculated as \(|\mathbf{v}| = \sqrt{a^2 + b^2}\).

This formula comes from the Pythagorean theorem, as it essentially measures the hypotenuse of a right-angled triangle. Keep in mind that the magnitude is always a non-negative number. Even if a vector has negative components, the squared values are positive, ensuring a positive magnitude.

Vector addition is a fundamental operation in vector arithmetic. Simply put, it involves adding together the corresponding components of the given vectors. Suppose you have two vectors, \(\mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j}\) and \(\mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j}\). The sum \(\mathbf{a} + \mathbf{b}\) is given by:

• \(\mathbf{a} + \mathbf{b} = (a_1 + b_1) \mathbf{i} + (a_2 + b_2) \mathbf{j}\)

This results in a new vector whose components are the sums of the respective components of the original vectors.
For instance, in our exercise, \(\mathbf{a} = -\mathbf{i} - \mathbf{j}\) and \(\mathbf{b} = 2\mathbf{i} + 2\mathbf{j}\). Adding them gives \(\mathbf{a} + \mathbf{b} = \mathbf{i} + \mathbf{j}\). This operation preserves the direction proportionally according to the individual components' directions.

Scalar multiplication involves multiplying a vector by a scalar (a constant real number). This operation scales the vector, changing its magnitude but not its direction. Suppose you have a vector \(\mathbf{v} = a\mathbf{i} + b\mathbf{j}\) and you multiply it by a scalar, say \(-2\). The result is:

• \(-2\mathbf{v} = -2(a\mathbf{i} + b\mathbf{j}) = (-2a)\mathbf{i} + (-2b)\mathbf{j}\)

Effectively, each component of the vector is multiplied by the scalar. The direction remains the same if the scalar is positive and reverses if the scalar is negative. For instance, in the step where \(-2 \mathbf{b}\) is calculated, the vector \(\mathbf{b}\) is multiplied by \(-2\), resulting in \(-4 \mathbf{i} - 4 \mathbf{j}\). This changes the vector's length but flips its direction.

Subtracting vectors is quite like adding them, but with a slight twist. You subtract one vector from another by subtracting each corresponding component. Given two vectors \(\mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j}\) and \(\mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j}\), the difference \(\mathbf{a} - \mathbf{b}\) is calculated as:

• \(\mathbf{a} - \mathbf{b} = (a_1 - b_1) \mathbf{i} + (a_2 - b_2) \mathbf{j}\)

This effectively creates a new vector that points from the tip of \(\mathbf{b}\) to the tip of \(\mathbf{a}\).
In the provided exercise, \(\mathbf{a} - \mathbf{b}\) was evaluated as \(-3 \mathbf{i} - 3 \mathbf{j}\). This operation is essential for determining relative positions and movements in space. Just remember to handle each component carefully, ensuring the correct vector gets subtracted.