Exercise 2: Dot Product, Predicting the Angle, and Confirming It — Possible Solution
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GIVEN
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c = [6, 8]
d = [-2, 1]

STEP 1: THE DOT PRODUCT
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c . d = (6 x -2) + (8 x 1) = -12 + 8 = -4

STEP 2: PREDICTING THE ANGLE CATEGORY FROM THE SIGN ALONE
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Per this chapter's own sign table, a negative dot product means the
angle between the two vectors is greater than 90 degrees. Since c . d
= -4 is negative, the prediction is: theta > 90 degrees.

STEP 3: COMPUTING THE ACTUAL ANGLE
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|c| = sqrt(6^2 + 8^2) = sqrt(100) = 10
|d| = sqrt((-2)^2 + 1^2) = sqrt(5) ~= 2.236

cos(theta) = (c . d) / (|c| |d|) = -4 / (10 x 2.236) = -4 / 22.36
           ~= -0.1789

theta = arccos(-0.1789) ~= 100.3 degrees

STEP 4: CONFIRMING THE PREDICTION
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100.3 degrees is indeed greater than 90 degrees, confirming the sign-
based prediction from Step 2 without needing the full calculation to
already know which side of 90 degrees the answer would land on.

WHY THIS WORKS AS AN ANSWER
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It uses the negative sign of the dot product to predict the angle
category immediately, per this chapter's own dot-product/angle
relationship, then carries out the full cos(theta) = (a.b)/(|a||b|)
calculation to get the precise angle, and explicitly checks the final
number against the earlier prediction rather than computing the two
independently.