If vectors v ⃗ = 12 ı ^ v⃗=12î v ⃗ = 12 ı ^ and u ⃗ = 100 ȷ ^ u⃗=100ĵ u ⃗ = 100 ȷ ^ , calculate u ⃗ ⋅ v ⃗ u ⃗⋅v ⃗ u ⃗ ⋅ v ⃗ .

Let's try solving this problem. So in this problem, we're told that vectors 'v' is equal to 12i and 'u' is equal to 100j, calculate the dot product of u and v. Now, something that I notice here is we're now given I and j notation, and it's important to know how to solve these types of problems no matter what notation you're given. Now since I see that we have vector v dotted with vector u, I can go ahead and just dot the product the vectors that we have. So we can see that v is 12 I, and that's going to be the dot product with u, which is a 100 j. Now here's something that you should notice. When dealing with the dot product, need to multiply the like components, so the x components and the y components, and then add them together, or in this case, we would say the I and the j components. But notice something here. We have an I right there, and we have a j right here. We have 2 different components. So how exactly would we find this dot product? Well, if we have 2 completely different components, meaning we have only x's here and then only y's there, what that means is that these vectors are actually perpendicular to each other. And whenever the vectors are perpendicular their dot product is always going to be 0. And a way that you can think about this is to actually think about what these vectors look like in the bracket notation. So 12 I will be the same thing as an x component of 12 and a y component of 0, and a 100 j would be the same thing as an x component of 0 and a y component of a 100. And if you did the dot product where you multiply 12 and 0 that would give you 0, and you could take 0 and multiply by a 100 and that would also give you 0, and 0 plus 0 is just 0. So it really makes sense that the dot product ends up being 0 when we multiply 2 vectors of completely different components, which are perpendicular. So hope you found this video helpful. Thanks for watching.

8. Vectors

Dot Product

Trigonometry

8. Vectors

Dot Product

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Multiple Choice

If vectors $v⃗=12î$ and $u⃗=100ĵ$ , calculate $u ⃗⋅v ⃗$ .

$1200k̂$

$1200$

$0$

$100$

Verified step by step guidance

Understand the dot product formula: The dot product of two vectors $ \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k} $ and $ \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} + b_3 \mathbf{k} $ is given by $ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 $.

Identify the components of the vectors $ \mathbf{v} $ and $ \mathbf{u} $: $ \mathbf{v} = 12\mathbf{i} $ and $ \mathbf{u} = 100\mathbf{j} $. Note that $ \mathbf{v} $ has components $ (12, 0, 0) $ and $ \mathbf{u} $ has components $ (0, 100, 0) $.

Apply the dot product formula: Substitute the components into the dot product formula: $ \mathbf{v} \cdot \mathbf{u} = 12 \times 0 + 0 \times 100 + 0 \times 0 $.

Calculate each term: Compute each multiplication separately: $ 12 \times 0 = 0 $, $ 0 \times 100 = 0 $, and $ 0 \times 0 = 0 $.

Add the results: Sum the products obtained from each term: $ 0 + 0 + 0 = 0 $. This is the dot product of the vectors $ \mathbf{v} $ and $ \mathbf{u} $.