If vectors a ⃗ = 13 ı ^ a⃗=13î a ⃗ = 13 ı ^ , ⃗ b ⃗ = 5 ı ^ − 12 ȷ ^ ⃗b⃗=5î-12ĵ ⃗ b ⃗ = 5 ı ^ − 12 ȷ ^ , and c ⃗ = 24 ȷ ^ c⃗=24ĵ c ⃗ = 24 ȷ ^ , calculate b ⃗ ⋅ ( a ⃗ − c ⃗ ) b ⃗⋅(a ⃗-c ⃗) b ⃗ ⋅ ( a ⃗ − c ⃗ ) .

Let's give this problem a try. So here we're told if vector 'a' is equal to 13i, vector 'b' is equal to 5i-12j, and vector c is equal to 24 j, calculate vector b dotted with a minus c. Now to find this, so we have b, and we need to find the dot product with a minus c, and the way we can calculate this is by just really kind of writing out everything that we have to really understand what's going on. So I can see the vector b is 5 I minus 12 j, and what we're doing is taking vector b, and we're dotting it with vector a minus c. Well, I can see that vector a is 13i, and I can see that vector c is 24j. And if we subtract vectors a and c, that's going to give us this new vector, and what we can do is take that vector and dot it with the vector we just found. So let's go ahead and run the dot product. Well for the dot product, we're going to take this 5, and we're going to multiply it by 13, and 5 times 13 is 65. And now what we're going to do is we're going to multiply the y components, and the y components are going to be negative 12 and negative 24. Negative 12 times negative 24 is positive 288. So So for the dot product, I can add these 2 together and that's going to give me the result. 65 +288, turns out to be 353. So 353 is the result for this product, this dot product, and this subtraction here, and that is the solution to the problem. So hope you found this video helpful. Thanks for watching.

14. Vectors

Dot Product

Precalculus

14. Vectors

Dot Product

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Multiple Choice

If vectors $a⃗=13î$ , $⃗b⃗=5î-12ĵ$ , and $c⃗=24ĵ$ , calculate $b ⃗⋅(a ⃗-c ⃗)$ .

$353$

$132$

$65$

$-223$

Verified step by step guidance

First, understand the operation involved: the dot product. The dot product of two vectors is calculated by multiplying their corresponding components and then summing those products.

Next, calculate the vector subtraction a⃗ - c⃗. Given a⃗ = 13ı^ and c⃗ = 24ȷ^, subtract the components: (13ı^ - 0ȷ^) - (0ı^ + 24ȷ^) = 13ı^ - 24ȷ^.

Now, you have the vector a⃗ - c⃗ = 13ı^ - 24ȷ^. You need to find the dot product of b⃗ and (a⃗ - c⃗). Given b⃗ = 5ı^ - 12ȷ^, write the expression for the dot product: (5ı^ - 12ȷ^) ⋅ (13ı^ - 24ȷ^).

Calculate the dot product by multiplying the corresponding components: (5 * 13) + (-12 * -24).

Finally, sum the products from the previous step to get the result of the dot product.

5:4m

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Struggling with Precalculus?

If vectors a⃗=13ı^a⃗=13îa⃗=13ı^, ⃗b⃗=5ı^−12ȷ^⃗b⃗=5î-12ĵ⃗b⃗=5ı^−12ȷ^, and c⃗=24ȷ^c⃗=24ĵc⃗=24ȷ^, calculate b⃗⋅(a⃗−c⃗)b ⃗⋅(a ⃗-c ⃗)b⃗⋅(a⃗−c⃗).

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Dot Product practice set

If vectors $a⃗=13î$ , $⃗b⃗=5î-12ĵ$ , and $c⃗=24ĵ$ , calculate $b ⃗⋅(a ⃗-c ⃗)$ .