Find a. 𝐯 ·𝐮,|𝐯|,|𝐮|. b. the cosine of the angle between 𝐯 and 𝐮. c. the scalar component of 𝐮 i

Find a. 𝐯 ·𝐮,|𝐯|,|𝐮|. b. the cosine of the angle between 𝐯...

Find a. $\mathbf{v} \cdot \mathbf{u},|\mathbf{v}|,|\mathbf{u}|$. b. the cosine of the angle between $\mathbf{v}$ and $\mathbf{u}$. c. the scalar component of $\mathbf{u}$ in the direction of $\mathbf{v}$. d. the vector proju $\mathbf{u}$. $$\mathbf{v}=2 \mathbf{i}-4 \mathbf{j}+\sqrt{5} \mathbf{k}, \quad \mathbf{u}=-2 \mathbf{i}+4 \mathbf{j}-\sqrt{5} \mathbf{k}$$

Find
a. $\mathbf{v} \cdot \mathbf{u},|\mathbf{v}|,|\mathbf{u}|$.
b. the cosine of the angle between $\mathbf{v}$ and $\mathbf{u}$.
c. the scalar component of $\mathbf{u}$ in the direction of $\mathbf{v}$.
d. the vector proju $\mathbf{u}$.
$$\mathbf{v}=2 \mathbf{i}-4 \mathbf{j}+\sqrt{5} \mathbf{k}, \quad \mathbf{u}=-2 \mathbf{i}+4 \mathbf{j}-\sqrt{5} \mathbf{k}$$

Key Concepts

Vector Projection

The vector projection of one vector onto another is the vector equivalent of the scalar projection. It is obtained by multiplying the scalar projection by the unit vector in the direction of the vector onto which the projection is made. This results in a vector that points in the same direction as the target vector and represents the component of the original vector along that direction.

Scalar Projection

The scalar projection of one vector onto another is the length of the component of the first vector in the direction of the second vector. It is computed by dividing the dot product of the two vectors by the magnitude of the direction vector. This measure captures the extent to which one vector points in the same direction as another.

Cosine of the Angle Between Two Vectors

The cosine of the angle between two vectors is defined using the dot product and the magnitudes of the vectors. Specifically, it is calculated as the dot product divided by the product of the magnitudes. This relationship provides insight into how aligned the two vectors are.

Vector Magnitude

The magnitude or norm of a vector is a measure of its length. It is calculated as the square root of the sum of the squares of its components. This concept is fundamental in normalizing vectors and in computations involving distances and angles in vector spaces.

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers (vectors) and returns a single number. It is found by multiplying corresponding entries of the two vectors and then summing those products. This operation is key in determining the angle between vectors and checking for orthogonality.

Transcript

00:01 Hi, so in this video we're going to talk about different components of v and you.

00:06 So in this first part of the question, it is asking us to find the dot product of v and u.

00:13 So in order to do that, the first thing that we're going to want to do is just write out our gibbons.

00:19 So v is 2i minus 4j plus the square root of 5k, and u is negative 2i plus 4j minus the square root of 5k.

00:37 And we can write this respectively as 2, negative 4, 5, and negative 2, 4, and negative 5.

00:46 Now using what we, that, we can do our that product.

00:50 So we're going to take 2 and multiply it by negative 2.

00:54 And we're going to add that to negative 4 times positive 4 plus negative 5.

01:02 Times positive 5.

01:03 And we find that that's going to be negative 25.

01:07 And that is the dot product.

01:09 And that is the first part.

01:11 Now we're going to find v.

01:13 And to do that, all we're going to do is take the square root of our numbers that are squared.

01:20 So v is 2.

01:21 We're going to square that plus negative 4, and we're going to square that, plus 5, and we're going to square that.

01:28 And that's all under a square root and we find that v is going to be five.

01:36 And we're going to do the same thing for you.

01:40 And if you can use a little bit of intuition here, you're probably going to figure out, oh, it's going to be five as well.

01:47 And you would be correct.

01:49 It's also going to be five.

01:50 We just need to finish writing it out...

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