Find two unit vectors that make an angle of 60° with v = ‹3, 4›. Give your answers correct to three decimal places.
I'm not sure what to do here; I believe that I use to definition of a dot product (or maybe cross product) with relation to the trig function (cos for dot, sin for cross) but for whatever reason I can't get my answer to come out.
let such a vector be (a,b)
then (a,b)⋅(3,4) = |(a,b)||(3,4)|cos60°
3a + 4b = 5√(a^2 + b^2)(1/2)
6a+8b = 5√(a^2+b^2)
let a= 1 , any non-zero number will do
6+8b = 5√(1+b^2)
square both sides
36+96b +64 b^2 = 25 + 25b^2
39b^2 + 96b + 11 = 0
b = (-96 ± √7500)/48
= appr. -2.341 or -0.1205
so two vectors could be (1, -2.341) or (1, -.1205)
but you want a unit vector, so using the second point,
|(1,-.1205)| = 1.00723
giving us a unit vector of (1/1.00723 , -.1205/1.00723)
or appr. (.9928 , -.1196)
check:
(3,4).(.9928,-.1196) = 2.5
|(3,4)||(.9928,-.1196)cos60 = 2.5
reverse the direction of the above vector to get
(-.9928, .1196) for a second vector, or
let a = some other value and repeat the above steps
Hey there! Don't worry, I'm here to clown around and help you out with this question.
To find unit vectors that make an angle of 60° with vector v = ‹3, 4›, we can use a little bit of math and some humor, of course!
First, let's find the magnitude of vector v. Using the Pythagorean theorem,
|v| = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.
Now, let's find the first unit vector. We can divide vector v by its magnitude:
u₁ = (1/|v|) * v = (1/5) * ‹3, 4› = ‹3/5, 4/5›.
Next, let's find the second unit vector. Since vectors ‹3/5, 4/5› and ‹-3/5, -4/5› are opposite directions, we can use the second unit vector as ‹-3/5, -4/5›.
So, to sum it all up, the two unit vectors that make an angle of 60° with v = ‹3, 4› are approximately:
u₁ ≈ ‹0.600, 0.800›
u₂ ≈ ‹-0.600, -0.800›
Remember, I'm here to bring some laughter to math problems, even if they make us feel like juggling with frustration!
To find two unit vectors that make an angle of 60° with v = ‹3, 4›, we can use the fact that the dot product of two vectors is equal to the product of their magnitudes multiplied by the cosine of the angle between them.
Let's start by finding the magnitude of the vector v:
|v| = √(3^2 + 4^2) = √(9 + 16) = √25 = 5
Now, we want to find two unit vectors u1 and u2 that make an angle of 60° with v, which means the dot product between v and u1 (or u2) should be equal to the product of their magnitudes (which is 5) multiplied by the cosine of 60° (which is 0.5).
Let's set up the equation:
v · u = |v| * |u| * cos(θ)
(3, 4) · (x, y) = 5 * 1 * 0.5
3x + 4y = 10 * 0.5 = 5
Now, we need to solve for x and y.
We can choose any value for x and find the corresponding value for y. To simplify the calculations, let's choose x = 2.
Plugging in the values into the equation:
3(2) + 4y = 5
6 + 4y = 5
4y = 5 - 6
4y = -1
y = -1/4
So, one unit vector that makes an angle of 60° with v is u1 = ‹2, -1/4›.
Now, let's choose a different value for x and find the corresponding value for y. Let's choose x = -2.
Plugging in the values into the equation:
3(-2) + 4y = 5
-6 + 4y = 5
4y = 5 + 6
4y = 11
y = 11/4
So, another unit vector that makes an angle of 60° with v is u2 = ‹-2, 11/4›.
Therefore, the two unit vectors that make an angle of 60° with v = ‹3, 4› are:
u1 ≈ ‹2, -0.25›
u2 ≈ ‹-2, 2.75›
Please note that I have rounded the values to three decimal places.
To find two unit vectors that make an angle of 60° with vector v = ‹3, 4›, you can follow these steps:
1. Normalize vector v: Find the magnitude of v by using the Pythagorean theorem.
|v| = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5.
Dividing each component of v by its magnitude gives you the unit vector u = (1/5)‹3, 4› = ‹3/5, 4/5›.
2. Use the dot product to find the angle between u and any vector making an angle of 60° with it.
Let w = ‹x, y› be a vector making an angle of 60° with u.
The dot product of u and w is given by u · w = |u| |w| cos(theta), where theta is the angle between u and w.
Since both u and w are unit vectors, their magnitudes are both 1, so the equation becomes u · w = cos(theta).
Substituting the values, (3/5)(x) + (4/5)(y) = cos(60°) = 1/2.
3. Solve the equation from step 2 for the variables x and y:
(3/5)(x) + (4/5)(y) = 1/2.
Multiply through by 5 to clear the fractions: 3x + 4y = 5/2.
Rearrange the equation to solve for one variable: y = (5/2 - 3x)/4.
4. Choose a value for x and calculate the corresponding value for y:
Let's choose x = 0:
y = (5/2 - 3(0))/4 = (5/2)/4 = 5/8.
So, one possible vector w is w = ‹0, 5/8›.
5. To find another vector making an angle of 60° with u, you can choose a different value for x and calculate the corresponding y value.
Let's choose x = 1:
y = (5/2 - 3(1))/4 = (5/2 - 3)/4 = -1/8.
So, another possible vector w is w = ‹1, -1/8›.
Therefore, two unit vectors making an angle of 60° with v = ‹3, 4› are ‹0, 5/8› and ‹1, -1/8›.