Use graph paper to find the resultant of each set of...

Use graph paper to find the resultant of each set of displacements.
$6 \overline{0} \mathrm{~km}$ due south, then $9 \overline{0} \mathrm{~km}$ at $15^{\circ}$ north of west, and then $75 \mathrm{~km}$ at $45^{\circ}$ north of east

Key Concepts

Vector Displacement

Displacement is a vector quantity that represents a change in position with both magnitude and direction. It is used to indicate not just how far an object has moved, but in which direction relative to a reference frame.

Vector Addition

Vector addition is the process of combining two or more vectors to obtain a single resultant vector. This operation takes into account both the magnitudes and directions of the individual vectors, leading to a net effect that may differ from that of any single vector.

Graphical Method for Vector Addition

The graphical method involves drawing vectors to scale on graph paper, placing them tip-to-tail, and then drawing a vector from the start point to the end point to represent the resultant. This method is particularly useful for visualizing the combined effect of multiple displacement vectors.

Vector Components

Resolving vectors into components involves breaking them down into perpendicular parts, typically along horizontal (x-axis) and vertical (y-axis) directions. By using trigonometric functions such as sine and cosine, each vector can be expressed in terms of these directional contributions, facilitating easier computation of the resultant vector.

Transcript

00:02 So in this example, they want us to find the resultant from the set of vectors given.

00:07 Now, this graph isn't drawn to scale, but it should serve its purpose for what we need it for.

00:14 So they tell us at first, we're going to travel due south 60 kilometers, and we'll call that vector a.

00:27 And then we're going to go 90 kilometers at 15 degrees north of west.

00:35 And we'll call that vector b.

00:40 And then from there, we're going to go 75 kilometers at 45 degrees north of east, which we'll call vector c.

00:51 So they want to know what the resultant is.

00:56 So in order for us to find the resultant and to find the magnitude of the resultant, we need to find the components, the x and y components of vectors a, b, and c.

01:06 So the components for a, we know that if we look at vector a, it's not going anywhere in the x direction.

01:17 So that will just be zero.

01:19 And the y direction, it's going the negative y direction.

01:24 So we know it's going to be negative.

01:26 We know the magnitude is 60 kilometers.

01:30 So it's going negative 60 kilometers in the y direction.

01:37 For b, we know it's going, or we're going to have to multiply the magnitude, which is 90 times cosine of 15 degrees to find its x component.

01:51 And we're going to need to multiply the magnitude by sign of 15 degrees to find its y component...

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