# What is the angle between #<9,-5,1 > # and #< -7,4,2 >#?

The angle is

The dot product is

So,

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To find the angle between two vectors ( \mathbf{u} ) and ( \mathbf{v} ), you can use the formula:

[ \cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| |\mathbf{v}|} ]

Where:

- ( \theta ) is the angle between the two vectors.
- ( \mathbf{u} \cdot \mathbf{v} ) is the dot product of the two vectors.
- ( |\mathbf{u}| ) and ( |\mathbf{v}| ) are the magnitudes (or lengths) of the vectors.

Given the vectors ( \mathbf{u} = \langle 9, -5, 1 \rangle ) and ( \mathbf{v} = \langle -7, 4, 2 \rangle ), you can calculate the dot product and the magnitudes:

[ \mathbf{u} \cdot \mathbf{v} = (9 \times -7) + (-5 \times 4) + (1 \times 2) ] [ |\mathbf{u}| = \sqrt{9^2 + (-5)^2 + 1^2} ] [ |\mathbf{v}| = \sqrt{(-7)^2 + 4^2 + 2^2} ]

After computing these values, substitute them into the formula to find ( \theta ), the angle between the two vectors.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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