Gurlll!!! open this file and summarize this notes and make me understand we are writing on it next😭 and if there are any questions solve them in a layman's way I I need to pass

Vectors are quantities that have both magnitude (size or length) and direction. They are often represented by arrows.

• The magnitude of a vector $\mathbf{a}$ is its length, written as $||\mathbf{a}||$.
• A unit vector is a vector with a magnitude of 1.
• Vectors can be added, subtracted, and multiplied by a scalar (a regular number).
• In 3D, a vector $\mathbf{a}$ can be written in component form as $\mathbf{a} = a_x\mathbf{i} + a_y\mathbf{j} + a_z\mathbf{k}$. Its magnitude is $||\mathbf{a}|| = \sqrt{a_x^2 + a_y^2 + a_z^2}$.
• The dot product $\mathbf{a} \cdot \mathbf{b}$ results in a scalar (a number). It's used to find the angle between two vectors or to check if they are perpendicular (if $\mathbf{a} \cdot \mathbf{b} = 0$).
• The cross product $\mathbf{a} \times \mathbf{b}$ results in a new vector that is perpendicular to both $\mathbf{a}$ and $\mathbf{b}$. Its magnitude equals the area of the parallelogram formed by $\mathbf{a}$ and $\mathbf{b}$. It's used to check if vectors are parallel (if $\mathbf{a} \times \mathbf{b} = \mathbf{0}$).
• The scalar triple product $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$ gives the volume of the parallelepiped formed by the three vectors. If this product is zero, the vectors are coplanar (they lie on the same plane).

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Example 1:
Given $\mathbf{a} = 2\mathbf{i} - 3\mathbf{j} + \mathbf{k}$ and $\mathbf{b} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k}$.

a) Find $\mathbf{a} + \mathbf{b}$

Step 1: Add the corresponding components of $\mathbf{a}$ and $\mathbf{b}$.

$$\mathbf{a} + \mathbf{b} = (2\mathbf{i} - 3\mathbf{j} + \mathbf{k}) + (\mathbf{i} + 2\mathbf{j} - 3\mathbf{k})$$ $$\mathbf{a} + \mathbf{b} = (2+1)\mathbf{i} + (-3+2)\mathbf{j} + (1-3)\mathbf{k}$$ $$\mathbf{a} + \mathbf{b} = 3\mathbf{i} - \mathbf{j} - 2\mathbf{k}$$

The sum of the vectors is $\boxed{\mathbf{3i} - \mathbf{j} - \mathbf{2k}}$.

b) Find $\mathbf{a} - \mathbf{b}$

Step 1: Subtract the corresponding components of $\mathbf{b}$ from $\mathbf{a}$.

$$\mathbf{a} - \mathbf{b} = (2\mathbf{i} - 3\mathbf{j} + \mathbf{k}) - (\mathbf{i} + 2\mathbf{j} - 3\mathbf{k})$$ $$\mathbf{a} - \mathbf{b} = (2-1)\mathbf{i} + (-3-2)\mathbf{j} + (1-(-3))\mathbf{k}$$ $$$$