Find the vectors whose lengths and directions are given. Try...

Find the vectors whose lengths and directions are given. Try to do the calculations without writing. $$\begin{array}{ll}{\text { Length }} & {\text { Direction }} \\ {\text { a. } 7} & {-\mathbf{j}} \\ {\text { b. } \sqrt{2}} & {-\frac{3}{5} \mathbf{i}-\frac{4}{5} \mathbf{k}} \\ {\text { c. } \frac{13}{12}} & {\frac{3}{13} \mathbf{i}-\frac{4}{13} \mathbf{j}-\frac{12}{13} \mathbf{k}} \\ {\text { d. } a>0} & {\frac{1}{\sqrt{2}} \mathbf{i}+\frac{1}{\sqrt{3}} \mathbf{j}-\frac{1}{\sqrt{6}} \mathbf{k}}\end{array}$$

Find the vectors whose lengths and directions are given. Try to do
the calculations without writing.
$$\begin{array}{ll}{\text { Length }} & {\text { Direction }} \\ {\text { a. } 7} & {-\mathbf{j}} \\ {\text { b. } \sqrt{2}} & {-\frac{3}{5} \mathbf{i}-\frac{4}{5} \mathbf{k}} \\ {\text { c. } \frac{13}{12}} & {\frac{3}{13} \mathbf{i}-\frac{4}{13} \mathbf{j}-\frac{12}{13} \mathbf{k}} \\ {\text { d. } a>0} & {\frac{1}{\sqrt{2}} \mathbf{i}+\frac{1}{\sqrt{3}} \mathbf{j}-\frac{1}{\sqrt{6}} \mathbf{k}}\end{array}$$

Key Concepts

Magnitude of a Vector

The magnitude of a vector, also known as its length, represents the size of the vector and is calculated using the square root of the sum of the squares of its components. It quantifies the distance represented by the vector regardless of its direction.

Direction Cosines

Direction cosines are the cosines of the angles between a vector and the coordinate axes. They are used to describe the orientation of a vector in space and help express the vector in component form based on its directional properties.

Unit Vector

A unit vector is a vector with a magnitude of one. It is used to specify direction alone without affecting magnitude and serves as a basis to construct other vectors by scaling with a scalar.

Scalar Multiplication

Scalar multiplication is the process of multiplying a vector by a real number (scalar) to change its magnitude while preserving or reversing its direction. This operation scales a unit vector to any desired length.

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