What is the projection of #< 3 , -7, 0># onto #< -1, -4 , 6 >#?
The vector projection is
The dot product is
Therefore,
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To find the projection of vector ( \mathbf{v} ) onto vector ( \mathbf{u} ), you use the formula:
[ \text{proj}_{\mathbf{u}}(\mathbf{v}) = \frac{\mathbf{v} \cdot \mathbf{u}}{| \mathbf{u} |^2} \mathbf{u} ]
Given ( \mathbf{v} = \langle 3, -7, 0 \rangle ) and ( \mathbf{u} = \langle -1, -4, 6 \rangle ),
[ \mathbf{v} \cdot \mathbf{u} = (3 \cdot -1) + (-7 \cdot -4) + (0 \cdot 6) = -3 + 28 + 0 = 25 ]
[ | \mathbf{u} | = \sqrt{(-1)^2 + (-4)^2 + 6^2} = \sqrt{1 + 16 + 36} = \sqrt{53} ]
[ \text{proj}_{\mathbf{u}}(\mathbf{v}) = \frac{25}{53} \langle -1, -4, 6 \rangle = \langle \frac{-25}{53}, \frac{-100}{53}, \frac{150}{53} \rangle ]
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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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- A triangle has sides A, B, and C. The angle between sides A and B is #(pi)/2#. If side C has a length of #12 # and the angle between sides B and C is #pi/12#, what is the length of side A?
- A triangle has sides A, B, and C. Sides A and B have lengths of 11 and 7, respectively. The angle between A and C is #(7pi)/24# and the angle between B and C is # (13pi)/24#. What is the area of the triangle?
- How do you solve the triangle given α = 15.6 degrees, b = 10.25, and c = 5.5?
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