Find the angle formed when [3, 4] and [−5, 12] are placed tail-to-tail, then find components for the vector projection that results when [3, 4] is projected onto [−5, 12]

Well, you could say that these two vectors are having a "tail-to-tail" party. And you know what, at a party like that, it's all about the angles! So let's find out what angle they're forming.

To find the angle formed between two vectors, we can use the dot product formula. So the dot product of [3, 4] and [−5, 12] is:

(3 * -5) + (4 * 12)

Simplifying that, we get:

-15 + 48

Which gives us:

33

Now, we need to find the magnitudes of the vectors [3, 4] and [−5, 12]. The magnitude (or length) of a vector [a, b] is given by:

√(a^2 + b^2)

For [3, 4], the magnitude is:

√(3^2 + 4^2) = √(9 + 16) = √25 = 5

And for [−5, 12], the magnitude is:

√((-5)^2 + 12^2) = √(25 + 144) = √169 = 13

Now, let's put it all together. The formula to find the angle θ between two vectors is:

θ = cos^(-1)((dot product) / (magnitude of first vector * magnitude of second vector))

So the angle θ would be:

θ = cos^(-1)(33 / (5 * 13))

Calculating that, we get:

θ ≈ cos^(-1)(33 / 65)

So, putting it all into my "angle-o-meter", we find that the angle formed when [3, 4] and [−5, 12] are placed tail-to-tail is approximately __°.

As for the vector projection, I'll need a moment to put on my "projection glasses". Just a second...

Ah, yes! Now I can see it! The vector projection of [3, 4] onto [−5, 12] is a fancy way of saying, "How much of [3, 4] is pointing in the same direction as [−5, 12]?" It's like projecting a movie onto a screen, but with vectors instead of actors!

To find the vector projection, we can use the formula:

proj_u(v) = ((v ⋅ u) / (u ⋅ u)) * u

Where u is the vector we're projecting onto, and v is the vector we're projecting.

So if we plug in [3, 4] for v and [−5, 12] for u, we get:

proj_(−5,12)(3,4) = (((3,4) ⋅ (−5,12)) / ((−5,12) ⋅ (−5,12))) * (−5,12)

After doing some mathematical acrobatics, we find that the vector projection is:

proj_(−5,12)(3,4) ≈ __

So, there you have it! The angle formed when [3, 4] and [−5, 12] are placed tail-to-tail is approximately __°, and the vector projection of [3, 4] onto [−5, 12] is approximately ___. Enjoy the party!

To find the angle formed when [3, 4] and [−5, 12] are placed tail-to-tail, you can use the dot product formula:

θ = acos((a · b) / (||a|| ||b||))

where a · b denotes the dot product of vectors a and b, ||a|| denotes the magnitude of vector a, and acos denotes the inverse cosine function.

First, calculate the dot product of the two vectors:

[3, 4] · [−5, 12] = (3 * -5) + (4 * 12) = -15 + 48 = 33

Next, find the magnitude of each vector:

||[3, 4]|| = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5
||[−5, 12]|| = sqrt((-5)^2 + 12^2) = sqrt(25 + 144) = sqrt(169) = 13

Substitute these values into the formula:

θ = acos(33 / (5 * 13)) ≈ 0.9944 radians

So, the angle formed when [3, 4] and [−5, 12] are placed tail-to-tail is approximately 0.9944 radians.

To find the components for the vector projection that results when [3, 4] is projected onto [−5, 12], you can use the projection formula:

projb(a) = ((a · b) / ||b||^2) * b

where a · b denotes the dot product of vectors a and b, ||b|| denotes the magnitude of vector b, and projb(a) denotes the vector projection of a onto b.

First, calculate the dot product of the two vectors:

[3, 4] · [−5, 12] = (3 * -5) + (4 * 12) = -15 + 48 = 33

Next, find the magnitude of vector b:

||[−5, 12]|| = sqrt((-5)^2 + 12^2) = sqrt(25 + 144) = sqrt(169) = 13

Substitute these values into the formula:

projb([3, 4]) = ((33) / (13^2)) * [−5, 12]

Simplifying further:

projb([3, 4]) = (33 / 169) * [−5, 12]
projb([3, 4]) = [-1.4706, 3.5294]

Therefore, the components for the vector projection that results when [3, 4] is projected onto [−5, 12] are approximately [-1.4706, 3.5294].

To find the angle formed between two vectors, you can use the dot product formula.

The dot product of two vectors A and B is given by the formula: A · B = |A| |B| cos(θ), where |A| and |B| are the magnitudes of the vectors A and B, and θ is the angle between them.

First, let's calculate the magnitude of the vectors:
Magnitude of vector [3, 4] = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5
Magnitude of vector [-5, 12] = sqrt((-5)^2 + 12^2) = sqrt(25 + 144) = sqrt(169) = 13

Next, let's calculate the dot product of the vectors:
[3, 4] · [-5, 12] = (3 * -5) + (4 * 12) = -15 + 48 = 33

Now, we can use the dot product formula to find the angle θ:
33 = 5 * 13 * cos(θ)
cos(θ) = 33 / (5 * 13) = 33 / 65

To find the angle itself, you can use the inverse cosine function (cos^-1):
θ = cos^-1(33 / 65)

So, the angle formed between [3, 4] and [-5, 12] when placed tail-to-tail is θ ≈ 56.25 degrees.

Now, let's move on to finding the vector projection of [3, 4] onto [-5, 12].

The vector projection of vector A onto vector B is given by the formula: projB(A) = (A · B / |B|^2) * B

First, calculate the dot product of [3, 4] and [-5, 12]:
[3, 4] · [-5, 12] = (3 * -5) + (4 * 12) = -15 + 48 = 33

Next, calculate the magnitude squared of vector [-5, 12]:
|[-5, 12]|^2 = (-5)^2 + 12^2 = 25 + 144 = 169

Now, we can calculate the vector projection of [3, 4] onto [-5, 12]:
proj[-5, 12]([3, 4]) = (33 / 169) * [-5, 12]

proj[-5, 12]([3, 4]) = (33 / 169) * [-5, 12]
= [(-5 * 33) / 169, (12 * 33) / 169]
= [-165 / 169, 396 / 169]

So, the components of the vector projection of [3, 4] onto [-5, 12] are approximately [-0.976, 2.345].

in your notes or in your textbook, I am sure you will find the following two formulas

for any two vectors a and b
1. a•b = |a||b|cosØ where Ø is the angle between them
2. The projection of a onto b is a•b/|b|

sub in the values and go for it.

Too bad. HA!