# 2.2 Coordinate systems and components of a vector  (Page 7/15)

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Check Your Understanding If the average velocity vector of the drone in the displacement in [link] is $\stackrel{\to }{u}=\left(15.0\stackrel{^}{i}+31.7\stackrel{^}{j}+2.5\stackrel{^}{k}\right)\text{m}\text{/}\text{s}$ , what is the magnitude of the drone’s velocity vector?

35.1 m/s = 126.4 km/h

## Summary

• Vectors are described in terms of their components in a coordinate system. In two dimensions (in a plane), vectors have two components. In three dimensions (in space), vectors have three components.
• A vector component of a vector is its part in an axis direction. The vector component is the product of the unit vector of an axis with its scalar component along this axis. A vector is the resultant of its vector components.
• Scalar components of a vector are differences of coordinates, where coordinates of the origin are subtracted from end point coordinates of a vector. In a rectangular system, the magnitude of a vector is the square root of the sum of the squares of its components.
• In a plane, the direction of a vector is given by an angle the vector has with the positive x -axis. This direction angle is measured counterclockwise. The scalar x -component of a vector can be expressed as the product of its magnitude with the cosine of its direction angle, and the scalar y -component can be expressed as the product of its magnitude with the sine of its direction angle.
• In a plane, there are two equivalent coordinate systems. The Cartesian coordinate system is defined by unit vectors $\stackrel{^}{i}$ and $\stackrel{^}{j}$ along the x -axis and the y -axis, respectively. The polar coordinate system is defined by the radial unit vector $\stackrel{^}{r}$ , which gives the direction from the origin, and a unit vector $\stackrel{^}{t}$ , which is perpendicular (orthogonal) to the radial direction.

## Conceptual questions

Give an example of a nonzero vector that has a component of zero.

a unit vector of the x -axis

Explain why a vector cannot have a component greater than its own magnitude.

If two vectors are equal, what can you say about their components?

They are equal.

If vectors $\stackrel{\to }{A}$ and $\stackrel{\to }{B}$ are orthogonal, what is the component of $\stackrel{\to }{B}$ along the direction of $\stackrel{\to }{A}$ ? What is the component of $\stackrel{\to }{A}$ along the direction of $\stackrel{\to }{B}$ ?

If one of the two components of a vector is not zero, can the magnitude of the other vector component of this vector be zero?

yes

If two vectors have the same magnitude, do their components have to be the same?

## Problems

Assuming the + x -axis is horizontal and points to the right, resolve the vectors given in the following figure to their scalar components and express them in vector component form.

a. $\stackrel{\to }{A}=+8.66\stackrel{^}{i}+5.00\stackrel{^}{j}$ , b. $\stackrel{\to }{B}=+30.09\stackrel{^}{i}+39.93\stackrel{^}{j}$ , c. $\stackrel{\to }{C}=+6.00\stackrel{^}{i}-10.39\stackrel{^}{j}$ , d. $\stackrel{\to }{D}=-15.97\stackrel{^}{i}+12.04\stackrel{^}{j}$ , f. $\stackrel{\to }{F}=-17.32\stackrel{^}{i}-10.00\stackrel{^}{j}$

Suppose you walk 18.0 m straight west and then 25.0 m straight north. How far are you from your starting point? What is your displacement vector? What is the direction of your displacement? Assume the + x -axis is horizontal to the right.

You drive 7.50 km in a straight line in a direction $15\text{°}$ east of north. (a) Find the distances you would have to drive straight east and then straight north to arrive at the same point. (b) Show that you still arrive at the same point if the east and north legs are reversed in order. Assume the + x -axis is to the east.

a. 1.94 km, 7.24 km; b. proof

A sledge is being pulled by two horses on a flat terrain. The net force on the sledge can be expressed in the Cartesian coordinate system as vector $\stackrel{\to }{F}=\left(-2980.0\stackrel{^}{i}+8200.0\stackrel{^}{j}\right)\text{N}$ , where $\stackrel{^}{i}$ and $\stackrel{^}{j}$ denote directions to the east and north, respectively. Find the magnitude and direction of the pull.

A trapper walks a 5.0-km straight-line distance from her cabin to the lake, as shown in the following figure. Determine the east and north components of her displacement vector. How many more kilometers would she have to walk if she walked along the component displacements? What is her displacement vector?

3.8 km east, 3.2 km north, 2.0 km, $\stackrel{\to }{D}=\left(3.8\stackrel{^}{i}+3.2\stackrel{^}{j}\right)\text{km}$

The polar coordinates of a point are $4\pi \text{/}3$ and 5.50 m. What are its Cartesian coordinates?

Two points in a plane have polar coordinates ${P}_{1}\left(2.500\phantom{\rule{0.2em}{0ex}}\text{m},\pi \text{/}6\right)$ and ${P}_{2}\left(3.800\phantom{\rule{0.2em}{0ex}}\text{m},2\pi \text{/}3\right)$ . Determine their Cartesian coordinates and the distance between them in the Cartesian coordinate system. Round the distance to a nearest centimeter.

${P}_{1}\left(2.165\phantom{\rule{0.2em}{0ex}}\text{m},1.250\phantom{\rule{0.2em}{0ex}}\text{m}\right)$ , ${P}_{2}\left(-1.900\phantom{\rule{0.2em}{0ex}}\text{m},3.290\phantom{\rule{0.2em}{0ex}}\text{m}\right)$ , 5.27 m

A chameleon is resting quietly on a lanai screen, waiting for an insect to come by. Assume the origin of a Cartesian coordinate system at the lower left-hand corner of the screen and the horizontal direction to the right as the + x -direction. If its coordinates are (2.000 m, 1.000 m), (a) how far is it from the corner of the screen? (b) What is its location in polar coordinates?

Two points in the Cartesian plane are A (2.00 m, −4.00 m) and B (−3.00 m, 3.00 m). Find the distance between them and their polar coordinates.

8.60 m, $A\left(2\sqrt{5}\phantom{\rule{0.2em}{0ex}}\text{m},0.647\pi \right)$ , $B\left(3\sqrt{2}\phantom{\rule{0.2em}{0ex}}\text{m},0.75\pi \right)$

A fly enters through an open window and zooms around the room. In a Cartesian coordinate system with three axes along three edges of the room, the fly changes its position from point b (4.0 m, 1.5 m, 2.5 m) to point e (1.0 m, 4.5 m, 0.5 m). Find the scalar components of the fly’s displacement vector and express its displacement vector in vector component form. What is its magnitude?

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