Given vectors u ⃗ u ⃗ u ⃗ and v ⃗ v ⃗ v ⃗ , sketch the resultant vector − 2 u ⃗ + 3 v ⃗ -2u ⃗+3v ⃗ − 2 u ⃗ + 3 v ⃗ .

Let's give this problem a try. So here we are given vectors u and v, and we're asked to sketch the resultant vector negative 2 u plus 3 v. So how can we go about doing this? Well, what I'm gonna do is I'm gonna do this step by step. I'm gonna start by dealing with this negative 2u. Now what I'm really going to do is focus on something even smaller and focus on this 2u right here, because 2u is just going to be scaling this vector or multiplying this vector u by a scalar. And since I can see that that scalar is 2, that means that we need to have twice the length of vector u. So this right here is vector u, if I have to draw vector 2 u, well vector 2 u would look something like this. So we can see it's 3 units up, so vector 2 u is going to be 6 units up. This would be the vector 2 u. Now notice we also have a negative sign, and because we have a negative sign that is going to cause the direction of this vector to point the opposite way. So vector negative 2 u would have the same length, but point in the opposite direction. So that's what our vector would look like. Now what we're asked to do is find negative 2 u plus 3 v. So what I'm going to do is take this negative 2 u and add it to 3 times v using the tip to tail method. Now first off, it's important that we know what vector 3v is going to look like, and 3v is just going to be 3 times the length of this vector. So I can see this vector is 2 units to the right and 1 unit up. Vector 3v would be 1, 2, 3, 4, 5, 6 units to the right and 3 units up. So if I start right here, I can go 1, 2, 3, 4, 5, 6 units to the right, and 123 units up, and this right here would be the vector 3 v. So now that we have vectors negative 2 u and vectors 3v, we've added those together, so what we just need to do is draw the resultant vector, which is going to go from the initial point of the first vector negative 2u to the terminal point of the last vector 3v and that is going to be our vector negative 2u+3v. Now I want to point out when I say 1st vector and last vector I'm talking about these 2 vectors we just connected. So that's what's going to give us this result vector which is the vector that we're looking for. So hope you found this video helpful, thanks for watching, and please let me know if you have any questions.

8. Vectors

Geometric Vectors

Trigonometry

8. Vectors

Geometric Vectors

Struggling with Trigonometry?

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Multiple Choice

Given vectors $u ⃗$ and $v ⃗$ , sketch the resultant vector $-2u ⃗+3v ⃗$ .

option a

option b

option c

option d

Verified step by step guidance

Identify the given vectors $ \vec{u} $ and $ \vec{v} $ from the first image. $ \vec{u} $ is a vertical vector pointing upwards, and $ \vec{v} $ is a horizontal vector pointing to the right.

To find $ -2\vec{u} $, reverse the direction of $ \vec{u} $ and double its length. This results in a vector pointing downwards with twice the length of $ \vec{u} $.

To find $ 3\vec{v} $, keep the direction of $ \vec{v} $ the same and triple its length. This results in a vector pointing to the right with three times the length of $ \vec{v} $.

To sketch the resultant vector $ -2\vec{u} + 3\vec{v} $, place the tail of $ 3\vec{v} $ at the head of $ -2\vec{u} $.

Draw the resultant vector from the tail of $ -2\vec{u} $ to the head of $ 3\vec{v} $. This vector represents $ -2\vec{u} + 3\vec{v} $.

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Struggling with Trigonometry?

Given vectors u⃗u ⃗u⃗ and v⃗v ⃗v⃗, sketch the resultant vector −2u⃗+3v⃗-2u ⃗+3v ⃗−2u⃗+3v⃗.

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Geometric Vectors practice set

Given vectors $u ⃗$ and $v ⃗$ , sketch the resultant vector $-2u ⃗+3v ⃗$ .