Find the unit vector in the direction of v⃗=12ı^−35ȷ^v⃗=12î-35ĵv⃗=12ı^−35ȷ^.

v ^ = 12 37 ı ^ − 35 37 ȷ ^ v̂=\frac{12}{37}î-\frac{35}{37}ĵ v ^ = 37 12 ı ^ − 37 35 ȷ ^

Find the unit vector in the direction of v ⃗ = 12 ı ^ − 35 ȷ ^ v⃗=12î-35ĵ v ⃗ = 12 ı ^ − 35 ȷ ^ .

Let's give this problem a try. So here we're asked to find the unit vector in the direction of 'v', where 'v' is equal to 12i minus 35j. Now, what I'm going to do is I'm going to write out what we need to find which is vector v hat. Now, to find this unit vector it's going to be the vector v divided by its magnitude. Now, the magnitude is not something that we're given so we're going to have to calculate it using this equation. Vector v's magnitude is going to be the square root of the x component squared plus the y component square. I can see that we're given both the x and the y component, so we can plug that into this equation. So we're going to have the square root of the I component which is 12 squared plus, and then we're going to have the y component or j which is 30 negative 35 squared, and it's negative because we have this minus sign here. Now 12 squared is a 144, and negative 35 squared, well, whenever you square a negative number it becomes positive, and it's going to be positive 1,220 5. Now adding these two numbers together underneath the square root will give you the square root of 1369. And And the square root of 1369 turns out to be 37. So this is what we end up getting for the magnitude of 'v'. So going back up to this equation, now that we found its magnitude, we can go ahead and take vector b, which is 12 I minus 35 j, and we can divide this by the magnitude 37. Now doing this, we're going to have 12 over 37 I splitting these 2 up into 2 separate fractions minus 35 divided by 37 j, and it turns out this fraction actually does not simplify down any farther. This is all that we get. So vector v hat is going to be 12 over 37 I minus 35 over 37 j that's the unit vector and that is the solution to the problem that's all there is to it. So that's that's how you can solve this problem. Hope you found this video helpful. Thanks for watching.

Find the unit vector in the direction of v ⃗ = 12 ı ^ − 35 ȷ ^ v⃗=12î-35ĵ v ⃗ = 12 ı ^ − 35 ȷ ^ .

v ^ = 12 37 ı ^ − 35 37 ȷ ^ v̂=\frac{12}{37}î-\frac{35}{37}ĵ v ^ = 37 12 ı ^ − 37 35 ȷ ^

v ^ = 37 ı ^ ^ v̂=37î̂ v ^ = 37 ı ^ ^

v ^ = 35 ı ^ − 12 ȷ ^ v̂=35î-12ĵ v ^ = 35 ı ^ − 12 ȷ ^

v ^ = 35 37 ı ^ − 12 37 ȷ ^ v̂=\frac{35}{37}î-\frac{12}{37}ĵ v ^ = 37 35 ı ^ − 37 12 ȷ ^

14. Vectors

Unit Vectors and i & j Notation

Precalculus

14. Vectors

Unit Vectors and i & j Notation

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Multiple Choice

Find the unit vector in the direction of $v⃗=12î-35ĵ$ .

$v̂=\frac{12}{37}î-\frac{35}{37}ĵ$

$v̂=37î̂$

$v̂=35î-12ĵ$

$v̂=\frac{35}{37}î-\frac{12}{37}ĵ$

Verified step by step guidance

First, understand that a unit vector is a vector with a magnitude of 1 that points in the same direction as the original vector. To find the unit vector, we need to divide the original vector by its magnitude.

Calculate the magnitude of the vector $ \mathbf{v} = 12\mathbf{i} - 35\mathbf{j} $. The magnitude $ ||\mathbf{v}|| $ is given by $ \sqrt{12^2 + (-35)^2} $.

Simplify the expression for the magnitude: $ ||\mathbf{v}|| = \sqrt{144 + 1225} = \sqrt{1369} $.

The magnitude $ ||\mathbf{v}|| $ is 37, so the unit vector $ \mathbf{v}^ $ is obtained by dividing each component of $ \mathbf{v} $ by 37.

Thus, the unit vector $ \mathbf{v}^ $ is $ \mathbf{v}^ = \frac{12}{37}\mathbf{i} - \frac{35}{37}\mathbf{j} $.

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