Find a magnitude and the unit vector of the following given that: a) a = 3i − 4j, b) b = i + 2j − k, c) c = i − 2k, d) d = 2j − 3k, e) e = 5i − 2j + k. 1) Find the magnitude. 2) Find the unit vector along the direction.

Here are the solutions to your questions:

1. Define the following and give five examples for each:

• Collinear Vectors: Vectors are collinear if they lie on the same line or are parallel to each other. This means one vector can be expressed as a scalar multiple of the other ($\vec{u} = k\vec{v}$ for some scalar $k$).

  • Examples:
    1. $\vec{u} = \vec{i}$, $\vec{v} = 3\vec{i}$
    2. $\vec{u} = 2\vec{j}$, $\vec{v} = -6\vec{j}$
    3. $\vec{u} = \vec{i} + \vec{j}$, $\vec{v} = 2\vec{i} + 2\vec{j}$
    4. $\vec{u} = (1, 2, 3)$, $\vec{v} = (3, 6, 9)$
    5. $\vec{u} = -\vec{k}$, $\vec{v} = 4\vec{k}$

• Non-collinear Vectors: Vectors are non-collinear if they do not lie on the same line and are not parallel. One vector cannot be expressed as a scalar multiple of the other.

  • Examples:
    1. $\vec{u} = \vec{i}$, $\vec{v} = \vec{j}$
    2. $\vec{u} = \vec{i} + \vec{j}$, $\vec{v} = \vec{i} - \vec{j}$
    3. $\vec{u} = (1, 0, 0)$, $\vec{v} = (0, 1, 0)$
    4. $\vec{u} = (1, 2, 0)$, $\vec{v} = (1, 0, 2)$
    5. $\vec{u} = \vec{i} + \vec{j} + \vec{k}$, $\vec{v} = \vec{i} + 2\vec{j} + 3\vec{k}$

2. Define the following and give five examples for each:

• Coplanar Vectors: Vectors are coplanar if they lie in the same plane. For three vectors, they are coplanar if their scalar triple product is zero ($\vec{a} \cdot (\vec{b} \times \vec{c}) = 0$). Any two vectors are always coplanar.

  • Examples:
    1. $\vec{u} = \vec{i}$, $\vec{v} = \vec{j}$, $\vec{w} = \vec{i} + \vec{j}$ (all in the $xy$-plane)
    2. $\vec{u} = (1, 0, 0)$, $\vec{v} = (0, 1, 0)$, $\vec{w} = (2, 3, 0)$ (all in the $xy$-plane)
    3. $\vec{u} = \vec{i}$, $\vec{v} = 2\vec{i}$, $\vec{w} = 3\vec{i}$ (all collinear, hence coplanar)
    4. $\vec{u} = (1, 1, 1)$, $\vec{v} = (2, 2, 2)$, $\vec{w} = (3, 3, 3)$ (all collinear, hence coplanar)
    5. $\vec{u} = (1, 0, 1)$, $\vec{v} = (0, 1, 1)$, $\vec{w} = (1, 1, 2)$ (since $\vec{w} = \vec{u} + \vec{v}$)

• Non-coplanar Vectors: Vectors are non-coplanar if they do not lie in the same plane. For three vectors, they are non-coplanar if their scalar triple product is non-zero ($\vec{a} \cdot (\vec{b} \times \vec{c}) \neq 0$).

  • Examples:
    1. $\vec{u} = \vec{i}$, $\vec{v} = \vec{j}$, $\vec{w} = \vec{k}$
    2. $\vec{u} = (1, 0, 0)$, $\vec{v} = (0, 1, 0)$, $\vec{w} = (0, 0, 1)$
    3. $\vec{u} = (1, 1, 0)$, $\vec{v} = (1, 0, 1)$, $\vec{w} = (0, 1, 1)$
    4. $\vec{u} = (1, 2, 3)$, $\vec{v} = (4, 5, 6)$, $\vec{w} = (7, 8, 10)$
    5. $\vec{u} = (1, 0, 0)$, $\vec{v} = (0, 2, 0)$, $\vec{w} = (0, 0, 3)$

3. Find the magnitude and the unit vector of the following vectors:

• a) $\vec{a} = 3\vec{i} - 4\vec{j}$