Final answer:
To find the angle between two vectors when a + b + c = 0 and |a| = 3, |b| = 5, and |c| = 7, we can use the dot product formula. Solving the equation -60 = 12(5) cos(θ), we find that cos(θ) = -1/2, which corresponds to angles of 120° or 240°. However, since none of the provided options match these angles, the correct answer is D. none of these.
Step-by-step explanation:
To find the angle between two vectors, we can use the dot product formula:
a · b = |a| |b| cos(θ)
Given that a + b + c = 0, we can rearrange it to a = -b - c. Substituting this into the dot product formula:
(-b - c) · b = |-b - c| |b| cos(θ)
Simplifying the dot product:
-(b · b) - (c · b) = |-b - c| |b| cos(θ)
Since |a| = 3, |b| = 5, and |c| = 7, the equation becomes:
-(5 · 5) - (7 · 5) = |-(-b - c)| |5| cos(θ)
Simplifying further:
-25 - 35 = |b + c| 5 cos(θ)
-60 = 12(5) cos(θ)
cos(θ) = -1/2
From the unit circle, we know that the angle whose cosine is -1/2 is 120° or 240°. However, since the options provided are 60°, 30°, 45°, and none of these, the correct answer is D. none of these.