Find the unit vector in the direction of a⃗=6ı^+3ȷ^a⃗=6î+3ĵa⃗=6ı^+3ȷ^.

a ^ = 2 √ 5 5 ı ^ + √ 5 5 ȷ ^ â=\frac{2\surd5}{5}î+\frac{\surd5}{5}ĵ a ^ = 5 2√5 ı ^ + 5 √5 ȷ ^

Find the unit vector in the direction of a ⃗ = 6 ı ^ + 3 ȷ ^ a⃗=6î+3ĵ a ⃗ = 6 ı ^ + 3 ȷ ^ .

Let's give this problem a try. So here we are given vector a, which is 6 I plus 3 j, and we are asked to find the unit vector in the direction of vector a. Now to find the unit vector for this specific vector that we have, what you need to do is take a, the vector, and then divide it by its magnitude. Now vector a is something we're given in the problem, but the magnitude is not something that we're given, so we're going to need to calculate it. Now to calculate the magnitude of a vector, you can use this equation. It's the square root of the x component squared plus the y component squared, and this will give you whatever the magnitude is. Now I can see that we have the x component, that's just going to be 6. We're going to have the square root of 6 squared plus 3 squared, and this is what's gonna go under the square root. Now 6 squared is 36 and 3 squared is 9. 36 plus 9 is 45. So the square root of 45 is what we end up getting, and it turns out that the square root actually simplifies. So if you were to break this down, you would actually get that this whole thing is equal to 3 times the square root of 5. In fact, you can plug both of these into your calculator and see that they will give you the same result. So 3 times the square root of 5 is the most simplified version of this radical that we get. So that's the magnitude of a. Now going back up to our unit vector here, I can take vector a which is given to us it's 6 I plus 3 j, and I can divide that by the magnitude, which is what we just calculated. So 3 times the square root of 5. So what this vector is going to be is 6 I divided by 3 times the square root of 5, plus 3 j divided by 3 times the square root of 5. Now it turns out that 3 can divide into 6 two times. So if I go ahead and cancel this 3 here, we're going to end up with just a 2 on top, and over here the threes are just gonna cancel completely. So what we're going to end up getting is 2 I over the square root of 5 plus j over the square root of 5. Now what I can do from here is I can actually rationalize this denominator by by multiplying the top and bottom by the square root of 5, because we don't typically want to have a square root in the denominator of a fraction. So what I can do is cancel the square roots down there, meaning that I'm going to end up with 2 times the square root of 5 divided by 5, and that's going to be I hat plus, and they're going to have the square root of 5 times j, and that's going to be divided by this quantity, which as you can see these squares will cancel here with this square here. So we're going to end up with just 5 on bottom as well. And then this right here is going to be what our vector looks like. So the unit vector a hat is going to be equal to 2 square root 5 divided by 5 in the I hat direction, plus square root of 5 divided by 5 in the j hat direction. And this right here should be the unit vector and the solution to the problem. So hope you found this video helpful. Thanks for watching.

Find the unit vector in the direction of a ⃗ = 6 ı ^ + 3 ȷ ^ a⃗=6î+3ĵ a ⃗ = 6 ı ^ + 3 ȷ ^ .

a ^ = 2 √ 5 5 ı ^ + √ 5 5 ȷ ^ â=\frac{2\surd5}{5}î+\frac{\surd5}{5}ĵ a ^ = 5 2√5 ı ^ + 5 √5 ȷ ^

a ^ = 3 √ 5 ȷ ^ â=3\surd5ĵ a ^ = 3√5 ȷ ^

a ^ = 2 √ 5 5 ı ^ − √ 5 5 ȷ ^ â=\frac{2\surd5}{5}î-\frac{\surd5}{5}ĵ a ^ = 5 2√5 ı ^ − 5 √5 ȷ ^

a ^ = 2 √ 5 5 ı ^ + 3 √ 5 5 ȷ ^ â=\frac{2\surd5}{5}î+\frac{3\surd5}{5}ĵ a ^ = 5 2√5 ı ^ + 5 3√5 ȷ ^

8. Vectors

Unit Vectors and i & j Notation

Trigonometry

8. Vectors

Unit Vectors and i & j Notation

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Multiple Choice

Find the unit vector in the direction of $a⃗=6î+3ĵ$ .

$â=3\surd5ĵ$

$â=\frac{2\surd5}{5}î-\frac{\surd5}{5}ĵ$

$â=\frac{2\surd5}{5}î+\frac{\surd5}{5}ĵ$

$â=\frac{2\surd5}{5}î+\frac{3\surd5}{5}ĵ$

Verified step by step guidance

First, understand that a unit vector in the direction of a given vector $ \vec{a} $ is obtained by dividing the vector by its magnitude. The given vector is $ \vec{a} = 6\hat{i} + 3\hat{j} $.

Calculate the magnitude of $ \vec{a} $ using the formula for the magnitude of a vector: $ ||\vec{a}|| = \sqrt{(6)^2 + (3)^2} $.

Simplify the expression for the magnitude: $ ||\vec{a}|| = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5} $.

Divide each component of the vector $ \vec{a} $ by its magnitude to find the unit vector: $ \hat{a} = \frac{6}{3\sqrt{5}}\hat{i} + \frac{3}{3\sqrt{5}}\hat{j} $.

Simplify the components of the unit vector: $ \hat{a} = \frac{2\sqrt{5}}{5}\hat{i} + \frac{\sqrt{5}}{5}\hat{j} $.

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