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A unit vector is a vector with a magnitude of 1. This means it indicates direction only, without extension or length.
Unit vectors are used in various calculations and representations in vector calculus to signify direction.
To find a unit vector in the same direction as a given vector \( \mathbf{a} \):

• Compute the magnitude of the vector \( \mathbf{a} \).
• Multiply the vector by the reciprocal of its magnitude.

For example, for vector \( \mathbf{a} = 7\mathbf{i} - 24\mathbf{j} \), its magnitude \( \| \mathbf{a} \| \) is 25.
Therefore, the unit vector \( \mathbf{u} \) is \( \frac{7}{25}\mathbf{i} - \frac{24}{25}\mathbf{j} \).
This vector maintains the direction of \( \mathbf{a} \) but reduces it to a magnitude of 1.

Vector magnitude refers to the length or size of the vector in space. To find the magnitude of a two-dimensional vector \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \), use the formula:

• \( \| \mathbf{a} \| = \sqrt{a_1^2 + a_2^2} \)

This formula is derived from the Pythagorean theorem and provides the "hypotenuse" or direct distance from the origin to the vector's endpoint.
For example, for vector \( \mathbf{a} = 7\mathbf{i} - 24\mathbf{j} \):

• \( \| \mathbf{a} \| = \sqrt{7^2 + (-24)^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \)

Thus, the magnitude of vector \( \mathbf{a} \) is 25, making it a vector of a substantial size before becoming a unit vector.

To find a vector directed in the opposite direction of a given vector, you simply multiply each component of the original vector by \(-1\).
This process effectively flips the arrowhead of the vector to point in the reverse direction.
In the context of unit vectors, when you have unit vector \( \mathbf{u} \) in one direction, the opposite unit vector \( \mathbf{v} \) is \(-\mathbf{u} \).
Using the earlier example, if \( \mathbf{u} = \frac{7}{25}\mathbf{i} - \frac{24}{25}\mathbf{j} \), then the vector in the opposite direction is:

• \( \mathbf{v} = -\mathbf{u} = -\frac{7}{25}\mathbf{i} + \frac{24}{25}\mathbf{j} \)

This opposite unit vector \( \mathbf{v} \) still maintains its unit length of 1, indicating it's just reversed in direction compared to \( \mathbf{u} \).