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What is the projection of [tex]\((2,6)\)[/tex] onto [tex]\((-1,5)\)[/tex]?

A. [tex]\(1.08(-1,5)\)[/tex]

B. [tex]\(0.70(-1,5)\)[/tex]

C. [tex]\(1.08(2,6)\)[/tex]

D. [tex]\(0.70(2,6)\)[/tex]


Sagot :

To find the projection of vector [tex]\((2, 6)\)[/tex] onto vector [tex]\((-1, 5)\)[/tex], we proceed through several steps:

1. Determine the Dot Product of the Vectors:
[tex]\[ \text{Dot product} = (2)(-1) + (6)(5) = -2 + 30 = 28 \][/tex]

2. Calculate the Magnitude Squared of the Second Vector:
[tex]\[ \text{Magnitude squared of } (-1, 5) = (-1)^2 + 5^2 = 1 + 25 = 26 \][/tex]

3. Compute the Projection Scalar:
[tex]\[ \text{Projection scalar} = \frac{\text{Dot product}}{\text{Magnitude squared}} = \frac{28}{26} = 1.07692307692 \][/tex]

4. Find the Projection by Multiplying the Scalar with the Second Vector:
[tex]\[ \text{Projection} = 1.07692307692 \times (-1, 5) \][/tex]

This gives us:
[tex]\[ \text{Projection} = (1.07692307692 \cdot -1, 1.07692307692 \cdot 5) = (-1.07692308, 5.38461538) \][/tex]

Therefore, the projection of [tex]\((2, 6)\)[/tex] onto [tex]\((-1, 5)\)[/tex] is represented by the scalar multiplying the vector [tex]\((-1, 5)\)[/tex]:

The correct answer is:
A. [tex]\(1.08(-1,5)\)[/tex]
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