Note: You can earn 10% partial credit for 2 - 4 correct answers, 60% partial cre

Question

Note: You can earn 10% partial credit for 2 - 4 correct answers, 60% partial credit for 5 - 6 correct answers, and 80% partial credit for 7 correct answers.

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Several unit vectors and in the xy-plane (not three-dimensional space) are shown in the figure.

Using the geometric definition of the dot product, are the following dot products positive, negative, or zero? You may assume that angles that look the same are the same.

Explanation / Answer

We know that dot product of two vectors is given by:-

a.b = |a||b|costheta

We can see the sign of dot product depends on costheta. If costheta is negative then dot product is negative and if cos theta is positive then dot product is also positive.

If 0 < theta < (90 deg.) => dot product is positive because cos(theta)>0

If (90 deg.) < theta < (180 deg.) => dot product is negative because cos(theta)<0

If theta = (90 deg.) => dot product is zero because cos(theta) = 0

If theta = (0 deg.) => dot product is positive because cos(theta) = 1

If theta = (180 deg.) => dot product is negative because cos(theta) = -1

we will use above rules to determine whetther dot product is positive or negative

1) s.t = positive (theta is less than 90)

2) r.s = positive (theta is less than 90)

3) t.u= positive (theta is less than 90)

4) r.u = negative (theta greater than 90)

5)n.e = 0 (theta = 90)

6)n.t = negative (theta greater than 90)

7)e.r = neagtive (theta greater than 90)

8) e.s = 0 (theta = 90)

Hope it Helps!!

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