If vectors ∣v⃗∣=12|v ⃗ |=12∣v⃗∣=12, ∣u⃗∣=100|u ⃗ |=100 ∣u⃗∣=100 and the angle between v⃗v ⃗v⃗ & u⃗u ⃗u⃗ is θ=π6\theta=\frac{\pi}{6}θ=6π, calculate v⃗⋅u⃗v ⃗⋅u ⃗v⃗⋅u⃗ .

600 3 600\sqrt3 600 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

If vectors ∣ v ⃗ ∣ = 12 |v ⃗ |=12 ∣ v ⃗∣ = 12 , ∣ u ⃗ ∣ = 100 |u ⃗ |=100 ∣ u ⃗∣ = 100 and the angle between v ⃗ v ⃗ v ⃗ & u ⃗ u ⃗ u ⃗ is θ = π 6 \theta=\frac{\pi}{6} θ = 6 π , calculate v ⃗ ⋅ u ⃗ v ⃗⋅u ⃗ v ⃗ ⋅ u ⃗ .

So let's try this problem where we're told if vector v, the magnitude of v is equal to 12, and the magnitude of vector u is a 100, and the angle between vectors v and u is pi over 6, calculate the dot product of v and u. Now, the way that we can solve this problem is using this equation that we've been talking about, which is that the dot product of v and u is equal to the magnitude of v times the magnitude of u times the cosine of the angle between the vectors. Now what I can see here is that we're trying to find the dot product of v and u, and we already have this angle given to us. So in essence, we're just gonna need to plug in the values that we have on the right side. So to do this, we'll have that vector v dotted with vector u is equal to the magnitude of v, which is 12, times the magnitude of u, which is a 100, times the cosine of the angle between vectors v and u, which is given to us as pi over 6 radians. Now if you look on the unit circle for the cosine of pi over 6, you're going to get that it's actually equal to the square root of 3 divided by 2. So that's what the cosine of pi over 6 is on the unit circle. Now what I can actually do to make things even more simple is I can take this 12, and I can divide it by 2. So I know that's just gonna give me 6. So we'll have 6 times a 100 times the square root of 3. Now 6 times a 100 is 600. So we end up with 600 times the square root of 3. Now at this point, you can use a calculator to go ahead and approximate this, and you should get about 1,039.23. So this right here is the result that you get for the dot product of vectors v and u. So both of these answers are correct. This is the exact answer. This is the approximation, And you could have just typed this into a calculator if you wanted to for this problem, but it's also important that you know how to do the unit circle and find the correct values because sometimes a calculator is actually not gonna help you when you need to find exact answers like the one that we have down here. But for this example, either one of these answers will be correct, and that is the solution to this problem. So hope you found this video helpful. Thanks for watching.

If vectors ∣ v ⃗ ∣ = 12 |v ⃗ |=12 ∣ v ⃗∣ = 12 , ∣ u ⃗ ∣ = 100 |u ⃗ |=100 ∣ u ⃗∣ = 100 and the angle between v ⃗ v ⃗ v ⃗ & u ⃗ u ⃗ u ⃗ is θ = π 6 \theta=\frac{\pi}{6} θ = 6 π , calculate v ⃗ ⋅ u ⃗ v ⃗⋅u ⃗ v ⃗ ⋅ u ⃗ .

600 3 600\sqrt3 600 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

300 3 300\sqrt3 300 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

8. Vectors

Dot Product

Trigonometry

8. Vectors

Dot Product

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Multiple Choice

If vectors $|v ⃗ |=12$ , $|u ⃗ |=100$ and the angle between $v ⃗$ & $u ⃗$ is $\theta=\frac{\pi}{6}$ , calculate $v ⃗⋅u ⃗$ .

$600$

$600\sqrt3$

$300\sqrt3$

$300$

Verified step by step guidance

Understand that the dot product of two vectors $ \vec{v} $ and $ \vec{u} $ is given by the formula $ \vec{v} \cdot \vec{u} = |\vec{v}| |\vec{u}| \cos(\theta) $, where $ |\vec{v}| $ and $ |\vec{u}| $ are the magnitudes of the vectors, and $ \theta $ is the angle between them.

Substitute the given values into the formula: $ |\vec{v}| = 12 $, $ |\vec{u}| = 100 $, and $ \theta = \frac{\pi}{6} $.

Calculate $ \cos(\theta) $ where $ \theta = \frac{\pi}{6} $. Recall that $ \cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2} $.

Substitute $ \cos(\theta) = \frac{\sqrt{3}}{2} $ into the dot product formula: $ \vec{v} \cdot \vec{u} = 12 \times 100 \times \frac{\sqrt{3}}{2} $.

Simplify the expression to find the dot product $ \vec{v} \cdot \vec{u} $.

5:40m

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