Evaluate the expression.

Question

Evaluate the expression.

r = <7, -3, -7>, v = <4, 6, -5>, w = <-5, -6, -3>

v ⋅ w

-41
<-20, -36, 15>
<-28, 18, -35>
-9

 

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A building has a ramp to its front doors to accommodate the handicapped. If the distance from the building to the end of the ramp is 19 feet and the height from the ground to the front doors is 4 feet, how long is the ramp? (Round to the nearest tenth.)

5.7 ft
18.6 ft
4.8 ft
19.4 ft

Answer 1

v dot w, generally means a vector dot product in this situation...

 

for two vectors v=<v_1,v_2,v_3> and w=<w_1,w_2,w_3>

 

their dot product is given by v•w=v_1*w_1+v_2*w_2+v_3*w_3

 

So 4*(-5)+6*(-6)+(-5)(-3) = ?? 

 

 

For the ramp problem...view it from  far away

 

top of building

 

Front door

      |

   4 feet

      |

GROUND       -19feet-          base of ramp leading up to door

 

We have a right triangle with the ramp being the hypotenuse, the building being the vertical leg and the ground being the horizontal leg...The lengths of the legs are 4 and 19 feet...so applying Pythagoras' Theorem...

 

|ramp|^2 = 4^2+19^2 = 377...so the ramp has length (absolute value, 1norm) equal to sqrt(377)...

 

An alternative quick approach is to recognize that the ramp has to be longer than the distance from the building to the base of the ramp (bc the hypotenuse is longer than each leg) and immediately select the only option that satisfies that condition!

 

 

Answer 2

v - w

 

This appears that we are subtracting elements of vectors.

 

<4, 6, -5> - <-5, -6, -3>  =

 

<(4 - (-5)) , (6 - (-6)) , ( -5 - (-3))> =

 

<9, 12, -2>

 

 

For the second part.

 

Draw a tall building with a ramp leaning on the building.  One end of the ramp touches the ground and the other end of the ramp is at doors height with the building.  The shape formed by the building and the ramp gets you a right triangle.

When we label the diagram based on the problem description, the horizontal leg is 19 ft and the vertical leg is 4 ft.  The hypotenuse of the triangle represents the length of the ramp.  To the find the length, we simply use Pythagorean theorem.

 

Length = √(19² + 4²)

           = √(361 + 16)

           = √(377)

           = 19.4

 

 

The ramp is 19.4 feet long.