Given vectors u ⃗ u ⃗ u ⃗ and v ⃗ v ⃗ v ⃗ , sketch the resultant vector 1 2 u ⃗ + v ⃗ \frac12u⃗+v⃗ 2 1 u ⃗ + v ⃗ .

Let's try solving this problem. So in this problem, we're told given vectors 'u' and 'v', sketch the resultant vector 1 half of 'u' plus 'v'. Okay. Now to solve this problem, I like to take these types of problems by the steps. And what I'm first going to do is figure out what vector 1 half of u is. Now you can see we have vector u given to us on this graph, and 1 half of u is going to scale this vector by a half. Now if we cut this vector in half, well, I can see the original vector is 1, 2 units to the right and 6 units up. So cutting this in half is going to be 1 unit to the right and 3 units up. So vector one half of u is going to look something like this. So this would be vector one half u, and what I need to do is add this to vector v. Now I can see that vector v is this vector right up here, and I can do this using the tip to tail method. So I'm going to redraw vector v, so it's down here, and connect these vectors tip to tail. So this is going to be vector v, and that's one half of u, and this is what they're going to look like connected. So all I need to do is draw the resultant vector, and the resultant vector is going to go from the initial point of our first vector to the terminal point of our second vector. So doing this is going to give us this vector right here, and this is the vector one half of u plus v. So that right there is the solution to our problem, and that is how you can solve these types of situations where you have to scale vectors and add them or subtract them. So hope you found this video helpful. Thanks for watching.

14. Vectors

Geometric Vectors

Precalculus

14. Vectors

Geometric Vectors

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Multiple Choice

Given vectors $u ⃗$ and $v ⃗$ , sketch the resultant vector $\frac12u⃗+v⃗$ .

option a

option b

option c

option d

Verified step by step guidance

Identify the given vectors $ \mathbf{u} $ and $ \mathbf{v} $ from the image. Note their direction and relative length on the grid.

To find $ \frac{1}{2} \mathbf{u} $, scale the vector $ \mathbf{u} $ by half its length while maintaining its direction. This involves reducing the length of $ \mathbf{u} $ by half.

Position the scaled vector $ \frac{1}{2} \mathbf{u} $ on the grid, starting from the origin or a reference point, ensuring it is half the length of $ \mathbf{u} $.

Add the vector $ \mathbf{v} $ to $ \frac{1}{2} \mathbf{u} $ using the tip-to-tail method. Place the tail of $ \mathbf{v} $ at the tip of $ \frac{1}{2} \mathbf{u} $.

Draw the resultant vector $ \frac{1}{2} \mathbf{u} + \mathbf{v} $ from the tail of $ \frac{1}{2} \mathbf{u} $ to the tip of $ \mathbf{v} $. This vector represents the sum of $ \frac{1}{2} \mathbf{u} $ and $ \mathbf{v} $.