If vectors ∣ a ⃗ ∣ = 3 |a⃗|=3 ∣ a ⃗∣ = 3 and ∣ b ⃗ ∣ = 7 |b⃗|=7 ∣ b ⃗∣ = 7 , and a ⃗ ⋅ b ⃗ = 14.85 a⃗\cdot b⃗=14.85 a ⃗ ⋅ b ⃗ = 14.85 , determine the angle between vectors a ⃗ a ⃗ a ⃗ and b ⃗ b ⃗ b ⃗ .

Let's give this problem a try. We're told if the magnitude of vectors a is equal to 3, and vector b is equal to 7, and the dot product of a and b is equal to 14.85, determine the angle between vectors a and b. So basically, we have these 2 random vectors a and b, we want to find the angle between them. Now, we know that the dot product of a and b is equal to the magnitude of a times the magnitude of b times the cosine of the angle between them. This is an equation we've talked about. Now the dot product between a and b, that's actually given to us as 14.85. So you can put 14.85 equal to the dot product of a and b, and that's going to equal the magnitude of a, which is 3, times the magnitude of b, which is 7, all multiplied by the cosine of the angle between the two vectors. So to find the angle, we just need to calculate theta. Now I can use some algebra to figure this out. So what I can do is take 3 times 7, and, by the way, 3 times 7 is 21. So I can take 21 and divide it on both sides of this equation, and that'll get the 3 times 7 and the 21 to cancel there. And then we'll have the that the cosine of theta is equal to 14.85 divided by 21. Now what I can do from here to find the angle is take the inverse cosine on both sides of this equation. The inverse cosine will cancel the cosine leaving me with just theta on the left side, and theta is the angle that we're looking for. So we'll have that theta is equal to the inverse cosine of the entire fraction that we have here. So it's gonna be the inverse cosine of 14.85 divided by 21. Now at this point you can type this into your calculator to see what this comes out to, and you should get an answer approximately equal to 45 degrees. So our angle theta is 45 degrees, and that is the angle between vectors A and B. So that's how you can solve this problem, hope you found this video helpful.

8. Vectors

Dot Product

Trigonometry

8. Vectors

Dot Product

Struggling with Trigonometry?

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Multiple Choice

If vectors $|a⃗|=3$ and $|b⃗|=7$ , and $a⃗\cdot b⃗=14.85$ , determine the angle between vectors $a ⃗$ and $b ⃗$ .

$21\degree$

$1.0\degree$

$45\degree$

$14.85\degree$

Verified step by step guidance

Start by recalling the formula for the dot product of two vectors: $ a \cdot b = |a| |b| \cos(\theta) $, where $ \theta $ is the angle between the vectors.

Substitute the given values into the formula: $ 14.85 = 3 \times 7 \times \cos(\theta) $.

Simplify the equation: $ 14.85 = 21 \cos(\theta) $.

Solve for $ \cos(\theta) $ by dividing both sides of the equation by 21: $ \cos(\theta) = \frac{14.85}{21} $.

Use the inverse cosine function to find $ \theta $: $ \theta = \cos^{-1}\left(\frac{14.85}{21}\right) $.

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Struggling with Trigonometry?

If vectors ∣a⃗∣=3|a⃗|=3∣a⃗∣=3 and ∣b⃗∣=7|b⃗|=7∣b⃗∣=7, and a⃗⋅b⃗=14.85a⃗\cdot b⃗=14.85a⃗⋅b⃗=14.85, determine the angle between vectors a⃗a ⃗a⃗ and b⃗b ⃗b⃗.

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Dot Product practice set

If vectors $|a⃗|=3$ and $|b⃗|=7$ , and $a⃗\cdot b⃗=14.85$ , determine the angle between vectors $a ⃗$ and $b ⃗$ .