True or false: If a ⃗ = ⟨ 3 , − 2 ⟩ a ⃗=⟨3,-2⟩ a ⃗ = ⟨ 3 , − 2 ⟩ and b ⃗ b ⃗ b ⃗ has initial point ( − 10 , 5 ) (-10,5) ( − 10 , 5 ) & terminal point ( − 7 , 3 ) (-7,3) ( − 7 , 3 ) , then a ⃗ = b ⃗ a ⃗=b ⃗ a ⃗ = b ⃗ .

Let's give this problem a try. Here we're asked true or false. If vector a is equal to 3 comma negative 2, and vector b has initial point negative 10, 5 and terminal point negative 7, 3, then vector a is equal to vector b. So this is a true or false question, and all we really need to do to solve this problem is figure out what vector b is. Because if it turns out that vector b is the same vector of as vector a, then this is a true statement. Otherwise, the statement is false. And the way that we can figure out what vector b is is using this equation. We're going to have x 2 minus x 1 comma y 2 minus y 1, where these x's and y's refer to the initial and terminal points that we are given. This is always the equation that you can use if you're given 2 points to represent your vector. Now what I'm going to go ahead and do is plug in the values. So we're going to have the terminal point for x, the x two, which I can see is negative 7. And then we're going to have this we're gonna subtract from this the initial point of x, which I can see is negative 10. Now this is going to be comma the difference in the y values, and the terminal y I can see is 3. Then I need to subtract off the initial y, which I can see is 5. So this is the vector that we end up getting. Now let's go ahead and do the math here. So So going to have negative 7 minus negative 10. This is the same thing as negative 7 plus 10, and negative 7 plus 10 is positive 3. This is going to be comma 3 minus 5, which is negative 2, and that's what we get for vector b. And notice how vector b is 3 negative 2, which is the same thing as vector a. So because we got the exact same vector, we would say that vector b is equal to vector a. So this statement is true, and that's how you can go about solving this problem. So hope you found this video helpful. Thanks for watching.

14. Vectors

Vectors in Component Form

Precalculus

14. Vectors

Vectors in Component Form

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Multiple Choice

True or false: If $a ⃗=⟨3,-2⟩$ and $b ⃗$ has initial point $(-10,5)$ & terminal point $(-7,3)$ , then $a ⃗=b ⃗$ .

True

False

Cannot be determined with the given information

Verified step by step guidance

First, understand that two vectors are equal if they have the same magnitude and direction. This means their components must be identical.

Given vector a⃗ = ⟨3, -2⟩, we need to find the components of vector b⃗.

To find the components of vector b⃗, use the formula for a vector given its initial point (x1, y1) and terminal point (x2, y2): b⃗ = ⟨x2 - x1, y2 - y1⟩.

Substitute the given points for b⃗: initial point (-10, 5) and terminal point (-7, 3). Calculate the components: b⃗ = ⟨-7 - (-10), 3 - 5⟩.

Simplify the components of b⃗: b⃗ = ⟨3, -2⟩. Now compare the components of a⃗ and b⃗. Since they are identical, a⃗ = b⃗, so the statement is True.

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Struggling with Precalculus?

True or false: If a⃗=⟨3,−2⟩a ⃗=⟨3,-2⟩a⃗=⟨3,−2⟩ and b⃗b ⃗b⃗ has initial point (−10,5)(-10,5)(−10,5) & terminal point (−7,3)(-7,3)(−7,3), then a⃗=b⃗a ⃗=b ⃗a⃗=b⃗.

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Vectors in Component Form practice set

True or false: If $a ⃗=⟨3,-2⟩$ and $b ⃗$ has initial point $(-10,5)$ & terminal point $(-7,3)$ , then $a ⃗=b ⃗$ .