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 $\mathbf{u}$. $$\mathbf{v}=-\mathbf{i}+\mathbf{j}, \quad \mathbf{u}=\sqrt{2} \mathbf{i}+\sqrt{3} \mathbf{j}+2 \mathbf{k}$$

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 $\mathbf{u}$.
$$\mathbf{v}=-\mathbf{i}+\mathbf{j}, \quad \mathbf{u}=\sqrt{2} \mathbf{i}+\sqrt{3} \mathbf{j}+2 \mathbf{k}$$

Key Concepts

Dot Product

The dot product is an operation between two vectors that multiplies corresponding components and sums the results. It is an essential concept in vector algebra because it provides a scalar that reflects the degree of alignment between the two vectors and is used in calculations involving angles and projections.

Vector Magnitude

The magnitude of a vector, also known as the norm, is a measure of its length and is calculated as the square root of the sum of the squares of its components. This fundamental concept is used to normalize vectors and to compute distances in Euclidean space.

Cosine of the Angle Between Vectors

The cosine of the angle between two vectors is obtained by dividing the dot product of the vectors by the product of their magnitudes. This value, which ranges from -1 to 1, indicates how aligned the two vectors are, where 1 means they are perfectly aligned, -1 means they are oppositely aligned, and 0 indicates orthogonality.

Scalar Projection

The scalar projection of one vector onto another represents the length of the component of the first vector in the direction of the second vector. It is computed as the dot product of the two vectors divided by the magnitude of the vector onto which the projection is being made, providing a measure of the ‘shadow’ of one vector in the direction of the other.

Vector Projection

The vector projection of one vector onto another is the vector that represents the component of the first vector in the direction of the second vector. It is calculated by multiplying the scalar projection by the unit vector in the direction of the second vector, yielding a vector that points in the same direction as the target vector with a magnitude equal to the scalar projection.

Transcript

00:01 Hi, so in this video we're going to be talking about what you can do with vectors.

00:04 The first thing we're going to talk about is the dot product.

00:07 All you have to do is multiply together the i values with the i value and the j value with the j value and the ks with the ks.

00:14 And so i'm just going to do that right now and i'm going to write it as such and you add that to the negative 1 times square root of 3.

00:24 And i'm not even going to write down the k values because we know those just cancel out to be 0.

00:29 And this equals negative square root 2 plus the square root of 3.

00:34 And that is our dot product.

00:36 Now for our v component, what we're going to do is take the square root of the whole thing, and we're going to square each of the vectors.

00:47 Like so.

00:49 And we find that our v is going to be square root 2.

00:54 And our u component is going to be the same thing.

00:57 Oh wow, that is a nasty looking square root.

01:02 Here we go.

01:03 That's even worse, maybe, but we're going to go with it.

01:06 So square root two squared plus square root three squared plus two squared, which equals the square root of nine, which comes up to me three.

01:19 So our answer for this part is 2, negative 2, negative square root 2 plus square root 3, square root two and three.

01:30 So we're going to use those values in these next parts...

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