Simple Nature - Light and Matter

r/The racing greyhound’s

velocity vector is in the direction

of its motion, i.e., tangent to its

curved path.

s/Example 65

A more compact notation is to write

∆r= (290 km)ˆx+ (230 km)ˆy,

where the vectorsˆx,ˆy, andˆz, called the unit vectors, are defined
as the vectors that have magnitude equal to 1 and directions lying
along thex,y, andzaxes. In speech, they are referred to as “x-hat,”
“y-hat,” and “z-hat.”

A slightly different, and harder to remember, version of this

notation is unfortunately more prevalent. In this version, the unit

vectors are calledˆi,ˆj, andˆk:

∆r= (290 km)ˆi+ (230 km)ˆj.

Applications to relative motion, momentum, and force

Vector addition is the correct way to generalize the one-dimensional

concept of adding velocities in relative motion, as shown in the fol-

lowing example:

Velocity vectors in relative motion example 65

.You wish to cross a river and arrive at a dock that is directly

across from you, but the river’s current will tend to carry you

downstream. To compensate, you must steer the boat at an an-

gle. Find the angleθ, given the magnitude,|vW L|, of the water’s

velocity relative to the land, and the maximum speed,|vBW|, of

which the boat is capable relative to the water.

.The boat’s velocity relative to the land equals the vector sum of

its velocity with respect to the water and the water’s velocity with

respect to the land,

vBL=vBW+vW L.

If the boat is to travel straight across the river, i.e., along they

axis, then we need to havevBL,x= 0. Thisxcomponent equals

the sum of thexcomponents of the other two vectors,

vBL,x=vBW,x+vWL,x,

or

0 =−|vBW|sinθ+|vW L|.

Solving forθ, we find

sinθ=|vW L|/|vBW|,

so

θ= sin−^1

|vW L|

vBW

.

Section 3.4 Motion in three dimensions 205