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Vector addition is a fundamental operation in mathematics involving vectors. Vectors are quantities that have both magnitude and direction, typically expressed in terms of their components. Adding vectors component-wise means that we add the corresponding components of each vector.

For instance, consider two vectors, \( \mathbf{u} = \mathbf{i} + \mathbf{j} \) and \( \mathbf{v} = 2 \mathbf{i} - 3 \mathbf{j} \). To find the result of \( \mathbf{u} + \mathbf{v} \), simply add \( \mathbf{i} \) components, and \( \mathbf{j} \) components separately:

• For \( \mathbf{i} \): \( 1 + 2 = 3 \).
• For \( \mathbf{j} \): \( 1 + (-3) = -2 \).

As a result, \( \mathbf{u} + \mathbf{v} = 3\mathbf{i} - 2\mathbf{j} \). This simple yet powerful operation is widely used in various applications, including physics and engineering.

Vector subtraction mirrors vector addition in terms of operations, but instead of adding components, we subtract them. Just like vector addition, subtraction is performed component-wise, aligning only similar components.

If we take the vectors \( \mathbf{u} = \mathbf{i} + \mathbf{j} \) and \( \mathbf{v} = 2 \mathbf{i} - 3 \mathbf{j} \), and compute \( \mathbf{u} - \mathbf{v} \), here's how you do it:

• Subtract the \( \mathbf{i} \) components: \( 1 - 2 = -1 \).
• Subtract the \( \mathbf{j} \) components: \( 1 - (-3) = 4 \).

Therefore, \( \mathbf{u} - \mathbf{v} = -\mathbf{i} + 4\mathbf{j} \). Subtraction can help determine the difference vector, which denotes how one vector offsets from another.

Scalar multiplication involves changing the magnitude of a vector without altering its direction. When a vector is multiplied by a scalar (a constant), each component of the vector is multiplied by that scalar.

For example, let's multiply vector \( \mathbf{u} = \mathbf{i} + \mathbf{j} \) by a scalar 2:

• Multiply each component: \( 2 \times \mathbf{i} = 2\mathbf{i} \) and \( 2 \times \mathbf{j} = 2\mathbf{j} \).

Hence, \( 2\mathbf{u} = 2\mathbf{i} + 2\mathbf{j} \).
Similarly, for \( \mathbf{v} = 2\mathbf{i} - 3\mathbf{j} \), if scaled by 2, you get:

• \( 2 \times 2\mathbf{i} = 4\mathbf{i} \).
• \( 2 \times (-3)\mathbf{j} = -6\mathbf{j} \).

Thus, \( 2\mathbf{v} = 4\mathbf{i} - 6\mathbf{j} \). Scalar multiplication is crucial in scaling forces, velocities, or other directional quantities without changing the direction.

The norm of a vector, also known as its length or magnitude, quantifies how long the vector is. It is derived using the Pythagorean theorem, which relates to the Euclidean distance formula in a plane.

For a vector \( \mathbf{u} = \mathbf{i} + \mathbf{j} \), the norm is calculated as follows:

• Sum the squares of its components: \( 1^2 + 1^2 = 2 \).
• Take the square root of this sum: \( \sqrt{2} \).

So, \( \|\mathbf{u}\| = \sqrt{2} \).
For \( \mathbf{v} = 2\mathbf{i} - 3\mathbf{j} \), its norm is:

• The squares: \( 2^2 + (-3)^2 = 4 + 9 = 13 \).
• The square root: \( \sqrt{13} \).

Thus, \( \|\mathbf{v}\| = \sqrt{13} \).
The norm helps in comparing vector lengths and is widely used in vector normalization, where vectors are converted to a unit length.