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.

Table of contents

Skip topic navigation

0. Fundamental Concepts of Algebra3h 29m
1. Equations and Inequalities3h 27m
2. Graphs1h 43m
3. Functions & Graphs2h 17m
4. Polynomial Functions1h 54m
5. Rational Functions1h 23m
6. Exponential and Logarithmic Functions2h 28m
7. Measuring Angles40m
8. Trigonometric Functions on Right Triangles2h 5m
9. Unit Circle1h 19m
10. Graphing Trigonometric Functions1h 19m
11. Inverse Trigonometric Functions and Basic Trig Equations1h 41m
12. Trigonometric Identities 2h 34m
13. Non-Right Triangles1h 38m
14. Vectors2h 25m
15. Polar Equations2h 5m
16. Parametric Equations1h 6m
17. Graphing Complex Numbers1h 7m
18. Systems of Equations and Matrices3h 6m
19. Conic Sections2h 36m
20. Sequences, Series & Induction1h 15m
21. Combinatorics and Probability1h 45m
22. Limits & Continuity1h 49m
23. Intro to Derivatives & Area Under the Curve2h 9m

14. Vectors

Geometric Vectors

Precalculus

14. Vectors

Geometric Vectors

Previous problemNext problem

Struggling with Precalculus?

Join thousands of students who trust us to help them ace their exams!Watch the first video

Multiple Choice

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

option a

option b

option c

option d

Previous problemNext problem

Verified step by step guidance

Identify the given vectors u⃗ and v⃗ from the first image. Vector u⃗ is pointing upwards, and vector v⃗ is pointing downwards to the right.

To find the resultant vector -2u⃗ + 3v⃗, first scale vector u⃗ by -2. This means reversing the direction of u⃗ and doubling its length.

Next, scale vector v⃗ by 3. This involves tripling the length of v⃗ while maintaining its direction.

Place the scaled vector -2u⃗ on the grid. Then, from the tip of -2u⃗, place the scaled vector 3v⃗.

The resultant vector -2u⃗ + 3v⃗ is the vector that starts at the tail of -2u⃗ and ends at the tip of 3v⃗. Draw this vector to complete the sketch.

3:48m

Watch next

Master Introduction to Vectors with a bite sized video explanation from Patrick

Start learning

Struggling with Precalculus?

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

Watch next

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