Find the component form of v given its magnitude and the...

Find the component form of $v$ given its magnitude and the angle it makes with the positive $x$ -axis. Then sketch v.
Magnitude $\|\mathbf{v}\|=2$
Angle $\mathbf{v}$ in the direction $\mathbf{i}+3 \mathbf{j}$

Key Concepts

Component Form of a Vector

Vectors can be expressed in terms of their components using standard basis vectors (like i and j in two dimensions). This representation simplifies many operations such as addition, scalar multiplication, and finding the vector's magnitude and direction.

Unit Vector

A unit vector is a vector with a magnitude of one and indicates direction. It is obtained by dividing a vector by its own magnitude, which is essential when you need to extract only the directional information from a given vector.

Scalar Multiplication

Scalar multiplication of a vector scales its magnitude while keeping its direction unchanged. When a vector’s direction is known and a specific magnitude is desired, the unit vector in that direction is multiplied by the scalar magnitude to obtain the desired vector.

Direction and Angle Decomposition

When a vector’s direction is specified, either by its angle with respect to the x-axis or by another vector indicating its direction, trigonometric functions or normalization techniques can be used to decompose this directional information into components. This process involves computing the cosine and sine of the direction angle or normalizing the given directional vector before scaling it to the required magnitude.

Transcript

00:01 Okay, let's say we have a vector that has a magnitude or length of two, and it goes in the same direction as i plus 3j, which is really the vector 1, because the number in front of i is 1, and the number of front of j is 3.

00:29 So in order to do this to find the component form of a vector with a magnitude of 2 and going in the direction as i plus 3j, we need to basically find the unit vector and then we're going to multiply it by the magnitude.

00:58 Let's do that.

00:59 So if we have i plus 3j, we're going to find its unit vector and then i'm just going to multiply it by the magnitude of 2.

01:07 To get to the vector that i want.

01:10 So how do you find a unit vector? you got to divide by the magnitude of that vector.

01:15 So we have 1 .3.

01:17 So to find its magnitude, we're going to do the square root of 1 squared plus 3 squared.

01:24 1 squared is 1.

01:26 3 squared is 9.

01:28 So we're looking at the square root of 10, which is 2 times 5, so eh, not really simplifiable.

01:34 So to get the unit vector for this, we are going to divide each one of these by the square root of 10.

01:46 And you should probably rationalize your denominator, which would result in us multiplying the top and the bottom by root 10.

02:01 So we end up with the square root of 10 over 10 and 3 root 10 over 10.

02:10 So that is the unit vector.

02:13 But now in order to get the component form of the specific vector we want, we need to multiply that by the magnitude of 2.

02:23 So i'm going to do two times this unit vector.

02:34 So if we do that, we're going to have, you can put this over 1 if it helps, 2 root 10 over 10, and 6 root 10 over 10.

02:47 And now we have some fractions that are simplifiable, so we can simplify the whole numbers that are not under the radical...

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