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 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 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.

8. Vectors

Dot Product

Trigonometry

8. Vectors

Dot Product

Struggling with Trigonometry?

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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 problem: We need to calculate the dot product of vector b⃗ with the result of the vector subtraction (a⃗ - c⃗).

Start by performing the vector subtraction a⃗ - c⃗. Given a⃗ = 13î and c⃗ = 24ĵ, subtract the components: (13î - 0ĵ) - (0î + 24ĵ) = 13î - 24ĵ.

Now, we have the vector a⃗ - c⃗ = 13î - 24ĵ. Next, we need to calculate the dot product b⃗ ⋅ (a⃗ - c⃗).

The vector b⃗ is given as 5î - 12ĵ. The dot product formula is: b⃗ ⋅ (a⃗ - c⃗) = (5î - 12ĵ) ⋅ (13î - 24ĵ).

Apply the dot product formula: (5 * 13) + (-12 * -24). Calculate each term separately and then sum them to find the dot product.

5:40m

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Struggling with Trigonometry?

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 $b ⃗⋅(a ⃗-c ⃗)$ .