What are (a) the x component and (b) the y component of a vector a⃗ in the x y plane if its dire

What are (a) the x component and (b) the y component of a...

What are (a) the $x$ component and (b) the $y$ component of a vector $\vec{a}$ in the $x y$ plane if its direction is $250^{\circ}$ counterclockwise from the positive direction of the $x$ axis and its magnitude is $7.3 \mathrm{~m} ?$

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Key Concepts

Vector Decomposition

Vector decomposition is the process of breaking a vector down into its orthogonal components (typically along the x and y axes). This allows for easier analysis and problem-solving, as many problems in physics and engineering involve handling these individual components independently.

Trigonometric Functions

Trigonometric functions, especially sine and cosine, are used to relate the magnitude of a vector to its components. Specifically, the x-component is found using the cosine of the given angle, and the y-component is found using the sine of the given angle, establishing a direct relationship between the angle and the vector’s projection on each axis.

Coordinate System Orientation

Understanding the orientation of a coordinate system is essential when decomposing vectors. Typically, the positive x-axis is used as the reference direction, and angles are measured counterclockwise from this axis. This convention determines the sign and magnitude of the resulting vector components.

Vector Magnitude

The magnitude of a vector represents its overall size or length and is a scalar quantity. When decomposing a vector, this magnitude is used in conjunction with trigonometric functions to determine how much of the vector lies along each coordinate axis, ensuring accurate representation of the original vector.

Transcript

00:01 In this example, we're going to be getting a little practice taking a vector from its magnitude and direction form and breaking it up into vector component form, so finding the x and y components.

00:13 So we have a vector.

00:16 Its angle is given as theta equals 250 degrees counterclockwise from the positive x -axis.

00:27 And its magnitude, so the magnitude of this vector v equals 7 .3 meters.

00:38 So we want to find its x and y components.

00:41 So first, we're going to need a little coordinate grid, so we'll put that over here.

00:49 So there's x, or that's y, rather.

00:57 Oh, my ruler disappeared.

01:00 And there we have the x -axis.

01:04 Ok, so we have positive x, positive y.

01:13 And we know, obviously, where the origin is.

01:16 So let's look at the angle first.

01:19 So we're starting here at 0, theta equals 0.