91Ó°ÊÓ

Vector addition is like putting arrows together. Vectors have directions and magnitudes. To add them, you add their corresponding components, which are the parts of the vector represented by numbers.
For example, consider vectors \( \mathbf{a} \) and \( \mathbf{b} \). If \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k} \) and \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k} \), the addition \( \mathbf{a} + \mathbf{b} \) results in another vector:

• \( (a_1 + b_1)\mathbf{i} \)
• \( (a_2 + b_2)\mathbf{j} \)
• \( (a_3 + b_3)\mathbf{k} \)

This means you simply add each component together to get the resultant vector.
So, for the example from the exercise, adding \( \mathbf{u} \) and \( \mathbf{v} \) resulted in \( 2\mathbf{i} - 2\mathbf{j} + 2\mathbf{k} \). Every new vector is essentially the result of moving one vector to the end of the other in a 'head-to-tail' fashion.

Scalar multiplication involves a single number, called a scalar, and a vector. When you multiply a vector by a scalar, each component of the vector is multiplied by that number.
For example, if you have a vector \( \mathbf{a} = 3\mathbf{i} - 4\mathbf{j} + \mathbf{k} \) and you multiply it by a scalar \( s \), you get:

• \( s \cdot 3\mathbf{i} = 3s\mathbf{i} \)
• \( s \cdot (-4)\mathbf{j} = -4s\mathbf{j} \)
• \( s \cdot 1\mathbf{k} = s\mathbf{k} \)

In the exercise, multiplication of vector \( \mathbf{u} \) by \(-2\) resulted in \(-2\mathbf{u} = -2(\mathbf{i} - 3\mathbf{j} + 2\mathbf{k}) \), which expanded to \(-2\mathbf{i} + 6\mathbf{j} - 4\mathbf{k} \). Scalar multiplication changes the size of a vector but not its direction, unless the scalar is negative, which reverses the direction.

A unit vector is a vector with a magnitude of 1. To create a unit vector from any vector, you divide it by its magnitude.
The formula for a unit vector \( \mathbf{u} \text{ from } \mathbf{v} \) is \( \mathbf{u} = \frac{1}{\|\mathbf{v}\|}\mathbf{v} \). This makes the vector have the same direction as \( \mathbf{v} \), but a length of just 1.
In the exercise, the unit vector of \( \mathbf{w} = 2\mathbf{i} + 2\mathbf{j} - 4\mathbf{k} \) was found by taking \( \frac{1}{2\sqrt{6}}(2\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}) \), resulting in \( \frac{1}{\sqrt{6}}(\mathbf{i} + \mathbf{j} - 2\mathbf{k}) \). Any vector can be turned into a unit vector, maintaining its own direction.

In calculus, vectors are used to represent quantities that have both magnitude and direction. Calculus with vectors often involves operations like differentiation and integration on vector functions.
Vector calculus is mainly used to study forces and motions in a multi-dimensional space. One interesting aspect is that it extends concepts like the derivative and integral from single-variable calculus to functions with multiple variables.
Some essential vector calculus operations include:

• Gradient: Measures the rate and direction of change in a scalar field.
• Divergence: Calculates the magnitude of a field's source or sink at a given point.
• Curl: Measures the twisting force over a vector field.

These concepts help in analyzing physical phenomena represented by vector fields, like fluid flow or electromagnetic fields. In our exercise, the focus isn't directly on calculus, but understanding vector operations lays the groundwork for studying vector calculus.