If #A = <2 ,2 ,5 >#, #B = <5 ,-7 ,8 ># and #C=A-B#, what is the angle between A and C?
The angle is
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The angle between vectors A and C can be found using the dot product formula. First, calculate vector C by subtracting vector B from vector A. Then, find the dot product of vectors A and C. Finally, use the dot product and the magnitudes of vectors A and C to calculate the angle between them using the formula:
[ \theta = \arccos\left(\frac{\mathbf{A} \cdot \mathbf{C}}{|\mathbf{A}| |\mathbf{C}|}\right) ]
Given vectors A and B, the components of vector C are obtained by subtracting the corresponding components of vector B from vector A:
[ C = A - B = \langle 2 - 5, 2 - (-7), 5 - 8 \rangle = \langle -3, 9, -3 \rangle ]
Next, calculate the dot product of vectors A and C:
[ \mathbf{A} \cdot \mathbf{C} = (2 \cdot -3) + (2 \cdot 9) + (5 \cdot -3) = -6 + 18 - 15 = -3 ]
Then, find the magnitudes of vectors A and C:
[ |\mathbf{A}| = \sqrt{2^2 + 2^2 + 5^2} = \sqrt{4 + 4 + 25} = \sqrt{33} ] [ |\mathbf{C}| = \sqrt{(-3)^2 + 9^2 + (-3)^2} = \sqrt{9 + 81 + 9} = \sqrt{99} ]
Finally, substitute these values into the formula to find the angle:
[ \theta = \arccos\left(\frac{-3}{\sqrt{33} \cdot \sqrt{99}}\right) ] [ \theta \approx \arccos\left(\frac{-3}{\sqrt{3267}}\right) ] [ \theta \approx \arccos\left(-0.1624\right) ] [ \theta \approx 98.84^\circ ]
Therefore, the angle between vectors A and C is approximately (98.84^\circ).
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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.
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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