# Vectors and direction/bearings.

• quantum_owl
That angle is 5 degrees. And, of course, you must remember that the components will be negative because they are to the left of the y-axis and below the x-axis.In summary, Jane walks a total of 400m due South, 250m in the direction S 85 degrees W, and 180m in the direction N 20 degrees E. Using the standard coordinate system where north is the positive y-direction and east is the positive x-direction, her displacement can be broken down into components of -250 sin(85)i- 250 cos(85)j and 180 cos(20)i+ 180 sin(20)j. These components can then be
quantum_owl

## Homework Statement

Jane walks 400m due South, then 250m in the direction S 85 degrees W, and finally 180m in the direction N 20 degrees E. Determine the magnitude and direction of her resulting displacement, relative to her starting point.

## Homework Equations

None, I don't think.

## The Attempt at a Solution

Well, I've gotten as far as trying to determine the magnitude of the second (250m) vector. My question is though, as she's heading south, does that mean that the vector is negative? Also, I'd also like to check with you very brainy, very lovely folks - the angle I've used for the
||AB||cos theta i etc is -95 degrees, as I've gone in the clockwise direction from the x-axis (90+90+85 = 265 degrees, then subtracting that from 360 degrees to give the negative 95 degrees).

I was sick when we did vectors in class, so I apologise if I've made some very glaring mistakes, but any help would be greatly appreciated.

There is no such thing as a "negative" (or "positive") vector. The indvidual components may be positive or negative. Strictly speaking whether they are positive or negative depends upon how YOU want to set up YOUR coordinate system but I suspect you are using the "standard"- north is the positive y-direction, east is the positive x-direction.

Also, I don't think you are "trying to determine the magnitude of the second (250m) vector." You are given that- it is 250. I think you are trying to determine the components.

In the standard coordinate system, going "400 m due south" would be <0, -400> or -400j. If you draw a line "south 85 degrees west" and then draw vertical and horizontal (north-south and east-west) lines you have a right triangle with angle 85 degrees (If you vertical line goes south from the first point and west east from the last point). Since she is going south and west, both components will be negative. The components will be <-250 sin(85), -250 cos(85)> or -250 sin(85)i- 250 cos(85)j. If your horizontal and vertical lines are west from the first point and north from the last, then the angle will 90- 85= 5 degrees but you also will reverse "opposite" and "near" sides. Since sin(85)= cos(5), that's the same thing.

If you are measuring angles counter-clockwise from the positive x-axis (east), which is the standard, then your angle will be 90+ 90+ 5= 185 degrees. And, in that standard, the x component is always cosine and y sine. Here, that would be <250 cos(185), 250 sin(185)>. Since cos(180+ 5)= -cos(5) and sin(180+ 5)= -sin(5) that's exactly what I had before.

Measuring "in the clockwise direction from the (positive) x-axis" would be 90+ 85, not 90+ 90+ 85. The first 90 takes you from east to south and then you have the "85 degrees west"

## 1. What are vectors and how are they used in science?

Vectors are quantities that have both magnitude and direction. They are used in science to represent physical quantities such as velocity, acceleration, and force. Vectors are often visualized as arrows, with the length of the arrow representing the magnitude and the direction of the arrow representing the direction.

## 2. How are vectors represented mathematically?

Vectors are typically represented mathematically using coordinates or components. In a Cartesian coordinate system, a vector is represented by an ordered pair (x, y) or a triplet (x, y, z). In component form, a vector is represented by its x, y, and z components, often denoted as i, j, and k respectively.

## 3. What is the difference between a vector and a scalar?

A vector has both magnitude and direction, while a scalar has only magnitude. For example, velocity is a vector quantity as it includes both speed (magnitude) and direction, while speed is a scalar quantity as it only has magnitude.

## 4. How are vectors added or subtracted?

In order to add or subtract vectors, their components are added or subtracted respectively. For example, to add two vectors A and B, we add their x, y, and z components: A + B = (Ax + Bx, Ay + By, Az + Bz). The resulting vector is the sum or difference of the two original vectors.

## 5. How are direction/bearings represented and measured?

Direction or bearings can be represented in various ways, such as using angles or compass directions. In science, direction is often represented using angles measured from a reference direction, such as north or east. Angles can be measured in degrees, radians, or other units. Compass directions, such as north, east, south, and west, can also be used to represent direction or bearings.

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