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. $\quad \mathbf{v} \cdot \mathbf{u},|\mathbf{v}|,|\mathbf{u}|$ 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 proje $\mathbf{u}$. $$\mathbf{v}=(3 / 5) \mathbf{i}+(4 / 5) \mathbf{k}, \quad \mathbf{u}=5 \mathbf{i}+12 \mathbf{j}$$

Find
a. $\quad \mathbf{v} \cdot \mathbf{u},|\mathbf{v}|,|\mathbf{u}|$
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 proje $\mathbf{u}$.
$$\mathbf{v}=(3 / 5) \mathbf{i}+(4 / 5) \mathbf{k}, \quad \mathbf{u}=5 \mathbf{i}+12 \mathbf{j}$$

Key Concepts

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and summing the products. It is fundamental in understanding the relationship between two vectors and is used to compute projections and the cosine of the angle between them.

Magnitude of a Vector

The magnitude (or norm) of a vector is a measure of its length. It is calculated by taking the square root of the sum of the squares of its components. This measurement is crucial when normalizing vectors and calculating angles between vectors.

Cosine of the Angle Between Vectors

The cosine of the angle between two vectors is obtained by dividing their dot product by the product of their magnitudes. This ratio provides insight into the directional relationship between the vectors and is widely used in many applications, including projections and determining vector similarity.

Scalar Projection

The scalar projection of one vector onto another represents the length of the orthogonal projection of the first vector along the direction of the second vector, preserving the sign to indicate the direction along the line defined by the second vector. It is calculated by multiplying the magnitude of the first vector by the cosine of the angle between the two vectors.

Vector Projection

The vector projection of one vector onto another is a vector that represents the component of the first vector in the direction of the second vector. It is obtained by scaling the second vector by the scalar projection factor divided by the square of its magnitude, effectively 'projecting' the first vector onto the direction defined by the second.

Transcript

00:03 We're given two vectors, and we want to find a number of things going on with these vectors.

00:08 And the good part is the things we find in part a are going to help us figure out the answers in the other three parts.

00:15 So we're going to start with part a.

00:18 And the first thing they asked us to find was vector v dot with vector u.

00:23 So we want to do the dot product, where we're just taking the same components and multiplying them together and then adding those up.

00:30 So we will have the i components, three -fifths times five, plus the j components.

00:38 Well, there's no j -component for v, so that's zero times the 12.

00:43 And then plus multiply the k components, that's four -fifths from v times there's no k component, so that's zero from vector u.

00:53 So when we do that, all the five is reduce and zero times anything is zero, so we end up with a dot product of three.

01:02 Second thing we need to find are the magnitude of both vectors.

01:06 The magnitude of vector v is going to be the square root of the x component, three -fifths squared, plus the y component, which is zero, plus the z component, oops, don't know why i'm doing a square root, plus the z component, which is four -fifths squared.

01:30 So when we square those, that's going to give us the square root of 9 .20, 25th plus 16, 25ths.

01:41 And when we add those together, that's 25, 25th, which is 1, and the square root of 1 is 1.

01:46 So that's the length of vector v.

01:49 And then we need the length of vector u.

01:52 Well, again, that's the square root of the components squared and added together.

01:58 So the square root of 5 squared plus 12 squared, and there's no z component, so i don't have to bother with that.

02:05 So that's the square root of 25 plus 144, which is 169, and the square root of 169 gives us 13.

02:18 So part a is done.

02:20 Part b is the cosine between of the angle that these two vectors form.

02:26 So the cosine of that angle, which we're going to call theta, equals the dot product 3, divided by the length of the two vectors multiplied together.

02:40 And that would be 13 times 1...

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