# If #A = <1 ,6 ,-8 >#, #B = <-9 ,4 ,1 ># and #C=A-B#, what is the angle between A and C?

The dot product divided by the magnitudes gives the cosine of the angle.

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To find the angle between vectors ( A ) and ( C ), first calculate the dot product of vectors ( A ) and ( C ), then use the dot product formula to find the angle.

Given that ( A = \langle 1, 6, -8 \rangle ), ( B = \langle -9, 4, 1 \rangle ), and ( C = A - B ), calculate ( C ) by subtracting the corresponding components of ( B ) from ( A ):

[ C = A - B = \langle 1, 6, -8 \rangle - \langle -9, 4, 1 \rangle = \langle 1 + 9, 6 - 4, -8 - 1 \rangle = \langle 10, 2, -9 \rangle ]

Next, calculate the dot product of ( A ) and ( C ):

[ A \cdot C = (1)(10) + (6)(2) + (-8)(-9) = 10 + 12 + 72 = 94 ]

Then, calculate the magnitudes of vectors ( A ) and ( C ):

[ |A| = \sqrt{1^2 + 6^2 + (-8)^2} = \sqrt{1 + 36 + 64} = \sqrt{101} ] [ |C| = \sqrt{10^2 + 2^2 + (-9)^2} = \sqrt{100 + 4 + 81} = \sqrt{185} ]

Now, use the dot product formula to find the angle ( \theta ) between ( A ) and ( C ):

[ \cos(\theta) = \frac{A \cdot C}{|A| \cdot |C|} = \frac{94}{\sqrt{101} \cdot \sqrt{185}} ]

[ \theta = \arccos\left(\frac{94}{\sqrt{101} \cdot \sqrt{185}}\right) ]

Finally, calculate the angle ( \theta ).

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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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