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 $\mathbf{u}$. $$\mathbf{v}=5 \mathbf{j}-3 \mathbf{k}, \quad \mathbf{u}=\mathbf{i}+\mathbf{j}+\mathbf{k}$$

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 $\mathbf{u}$.
$$\mathbf{v}=5 \mathbf{j}-3 \mathbf{k}, \quad \mathbf{u}=\mathbf{i}+\mathbf{j}+\mathbf{k}$$

Key Concepts

Vector Projection

The vector projection of one vector onto another provides a vector that represents the component of the first vector that lies in the direction of the second. This is obtained by multiplying the scalar projection by the unit vector in the direction of the second vector. Vector projections are useful in decomposing vectors and analyzing directional influences in various fields.

Scalar Projection

The scalar projection of one vector onto another is the length of the shadow that the first vector casts on the line defined by the second vector. It is computed by taking the dot product of the two vectors and dividing by the magnitude of the vector onto which the projection is made. This measure is used to determine how much of one vector lies in the direction of another.

Dot Product

The dot product is a fundamental operation in vector algebra that multiplies corresponding components of two vectors and sums the results. It provides a scalar measure of the extent to which two vectors point in the same direction. This concept is pivotal in determining angles between vectors and in projecting one vector onto another.

Vector Magnitude

The magnitude of a vector, often called its length or norm, measures the distance from the origin to the point represented by the vector. It is computed as the square root of the sum of the squares of its components. This concept is essential for normalizing vectors and in many applications across physics and engineering.

Cosine of the Angle Between Vectors

The cosine of the angle between two vectors is calculated by dividing their dot product by the product of their magnitudes. This value, ranging from -1 to 1, quantifies the directional similarity of the vectors. It is a critical component in understanding the geometric relationship between vectors in Euclidean space.

Transcript

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

00:05 The first thing that we're going to do is dot product.

00:08 So we're just going to multiply together all of our values that come first.

00:13 So this value goes with this value, this goes with this, this goes with this.

00:17 And we just add those all together.

00:21 We find that our answer is going to be 2 for our dot product.

00:28 Now for this next part, our v component, we're going to take our values that we have for our v vector, we're going to square them, and then we're going to add them all together and take the square root.

00:40 And then we will get, for this value, we end up getting the square root of 34.

00:50 And for u, we're going to do the exact same thing.

00:55 We just did for v, but we're going to use our u values.

00:59 It's quite some of the same thing.

01:00 For this one, as i'm sure you know, because one squared is just going to be one, so you get the square root of three.

01:07 So the answers for this question are two, square root of 34, and the square root of three.

01:13 Now, moving on, we're going to find the angle between v and u.

01:17 So in order to find cosine theta, we just need to use our dot product that we just found, which is represented by this highlighted part, and we know that is going to be two...

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