Find the unit vector that points in the same direction as...

Find the unit vector that points in the same direction as the given vector and express it in terms of the standard basis vectors. $$\begin{array}{ll}{\text { (a) } \mathbf{i}+\mathbf{j}} & {\text { (b) } \mathbf{i}+\mathbf{j}+\mathbf{k}} \\ {\text { (c) } 2 \mathbf{i}-\mathbf{k}} & {\text { (d) } 4 \mathbf{i}+6 \mathbf{j}-\mathbf{k}}\end{array}$$

Find the unit vector that points in the same direction as the given vector and express it in terms of the standard basis vectors.
$$\begin{array}{ll}{\text { (a) } \mathbf{i}+\mathbf{j}} & {\text { (b) } \mathbf{i}+\mathbf{j}+\mathbf{k}} \\ {\text { (c) } 2 \mathbf{i}-\mathbf{k}} & {\text { (d) } 4 \mathbf{i}+6 \mathbf{j}-\mathbf{k}}\end{array}$$

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Key Concepts

Magnitude of a Vector

The magnitude (or length) of a vector is the measure of its size and is computed using the Euclidean norm, which involves taking the square root of the sum of the squares of its components. This concept is essential for determining the scale of vectors and for normalizing them to obtain unit vectors.

Standard Basis Vectors

Standard basis vectors are a set of vectors that define the coordinate axes in a vector space, such as i, j, and k in three-dimensional space. Each vector in this set has a magnitude of 1 and points along one of the coordinate axes, providing a foundation for expressing any vector in terms of these directional components.

Unit Vector

A unit vector is a vector with a magnitude of 1. It is used to denote direction without consideration of magnitude and is usually derived by dividing a given vector by its magnitude. This concept is crucial in many areas of geometry and physics where the direction is needed but the scale is normalized.

Vector Normalization

Vector normalization is the process of converting a given vector into a unit vector. This is accomplished by dividing each component of the vector by its magnitude, thereby maintaining the direction while standardizing the length to one. This technique is widely used in various applications, including computer graphics and machine learning.

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