True or false: If a ⃗ = ⟨ 3 , 2 ⟩ a ⃗=⟨3,2⟩ a ⃗ = ⟨ 3 , 2 ⟩ and b ⃗ b ⃗ b ⃗ has initial point ( 3 , − 1 ) (3,-1) ( 3 , − 1 ) & terminal point ( 6 , 1 ) (6,1) ( 6 , 1 ) , then a ⃗ = b ⃗ a ⃗=b ⃗ a ⃗ = b ⃗ .

Let's try solving this problem. So here we're asked true or false. If vector a is equal to 3 comma 2, and vector b has initial point 3 negative 1, and terminal point 61, then vector a is equal to vector b. Okay. Now to solve this problem, all we really need to do is figure out what vector b is equal to. Because if we find vector b and it turns out to be the same as vector a, then this is a true statement. Otherwise, it's false. Now the way we can figure out what vector b is is recall this equation. Vector b is gonna be equal to x 2 minus x 1 comma y2 minus y 1. This refers to the initial and terminal points, and this is how you can find what a vector is if you're given the initial and terminal points. So what I can do is plug in the numbers to see what we get. Now I can see that x 2 is going to correspond to the terminal point that we have for our x. And I can see that that number is 6, and we're going to subtract off x 1, which I can see is going to correspond to the initial x that we have, which is 3. And that's going to be comma y 2, where y 2 is this terminal y value, which is 1 minus y 1, which is the initial y value negative 1. So all I need to do is this operation right here. So we're going to have 6 minus 3, which is 3 comma 1 minus negative one. Well, that's the same thing as 1+1, and 1+1 is 2. So vector b is equal to 3 comma 2. And notice how vector b is the same vector as vector a. We got 3 comma 2 for both of these. So what that means is that this statement is true. Vector a is equal to vector b, and that's how you can solve this problem. Hope you found this video helpful. Thanks for watching.

8. Vectors

Vectors in Component Form

Trigonometry

8. Vectors

Vectors in Component Form

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Multiple Choice

True or false: If $a ⃗=⟨3,2⟩$ and $b ⃗$ has initial point $(3,-1)$ & terminal point $(6,1)$ , then $a ⃗=b ⃗$ .

True

False

Cannot be determined with given information

Verified step by step guidance

Identify the components of vector $ \mathbf{a} $. Given $ \mathbf{a} = \langle 3, 2 \rangle $, the components are $ x_1 = 3 $ and $ y_1 = 2 $.

Determine the components of vector $ \mathbf{b} $. The initial point is $ (3, -1) $ and the terminal point is $ (6, 1) $. Calculate the components using the formula: $ x_2 = x_{\text{terminal}} - x_{\text{initial}} $ and $ y_2 = y_{\text{terminal}} - y_{\text{initial}} $.

Calculate the $ x $-component of $ \mathbf{b} $: $ x_2 = 6 - 3 = 3 $.

Calculate the $ y $-component of $ \mathbf{b} $: $ y_2 = 1 - (-1) = 2 $.

Compare the components of $ \mathbf{a} $ and $ \mathbf{b} $. If both $ x $-components and $ y $-components are equal, then $ \mathbf{a} = \mathbf{b} $.