Question text Let $$ \mathbf{a} = \langle 1,-3,4 \rangle \quad \text{and} \quad \mathbf{b} = \langle 0,5,5 \rangle. $$ The dot product $ \mathbf{a} \cdot \mathbf{b} $ is given by $ \mathbf{a} \cdot \mathbf{b} = $ Answer 1 Question 6[input]. The angle between $ \mathbf{a} $ and $ \mathbf{b} $ is Answer 2 Question 6[select: , zero, acute, a right angle, obtuse, a straight angle].

Question

BlackTom AI · Accepted Answer

The dot product of two vectors a and b is computed componentwise: a · b = a1b1 + a2b2 + a3b3. This value, together with the magnitudes |a| and |b|, determines the cosine of the angle θ between them by cos θ = (a · b)/(|a||b|). A positive dot product gives an acute angle, zero gives a right angle, and a negative dot product gives an obtuse angle.

Step-by-step calculation:
1. Compute the dot product a · b:
   a = ⟨1, −3, 4⟩, b = ⟨0, 5, 5⟩
   a · b = 1·0 + (−3)·5 + 4·5
         = 0 − 15 + 20
         = 5.
   This yields the first answer: 5.

2. Compute the magnitudes to find the angle:
   |a| = sqrt(1^2 + (−3)^2 + 4^2) = sqrt(1 + 9 + 16) = sqrt(26).
   |b| = sqrt(0^2 + 5^2 + 5^2) = sqrt(25 + 25) = sqrt(50) = 5√2.

3. Compute cos θ:
   cos θ = (a · b) / (|a||b|) = 5 / (sqrt(26) · sqrt(50)) = 5 / sqrt(1300).
   Simplify: sqrt(1300) = 10√13, so cos θ = 5 / (10√13) = 1 / (2√13) ≈ 0.1386.

4. Determine the angle type:
   Since cos θ ≈ 0.1386 > 0, the angle θ is between 0° and 90°, i.e., acute. (For reference, θ ≈ arccos(0.1386) ≈ 82.0°.)

Why the correct answers are correct:
- The arithmetic of the dot product directly gives 5, as shown.
- A positive dot product (here 5) implies cos θ > 0 and thus an acute angle.

Why other multiple-choice options are incorrect:
- zero: would require a · b = 0 (not true).
- a right angle: corresponds to dot product 0 (not true).
- obtuse: would require a · b < 0 (not true).
- a straight angle: would require θ = 180° and a · b = −|a||b| (not true).

Conclusion: The dot product is 5 and the angle between a and b is acute, matching the provided answers.

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