Question

In figure, the line segment represents a path from the point segment represents a path from the point with the position vector (5i + 3j)m to the Point with the location (16i + 12j) m. Point A is alone with this path a fraction f of the way to the destination.

a) Find the position vector of point A in terms of f.

b) Evaluate the expression from part a for f = 0.

c)Explain whether the result in part b is reasonable.

d) Evaluate the expression for f = 1.

e)Explain whether the results in part d is reasonable.

Short answer

1. The position vector of point A is $\left[\left(5+11f\right)i+\left(3+9f\right)j\right]$m.
2. For f = 0 we havethe position vector 5i + 3j m.
3. This is reasonable.
4. For f = 0 we have the position vector 16i + 12j m.
5. This is reasonable.

Step-by-step solution

Definition of vector

A vector is a quantity that has both a direction and a magnitude, and is used to determine the relative location of two points in space.

Formula

If vector A = a₁i + a₂j + a₃k and vector B = b₁i + b₂j + b₃k then the displacement of AB = (b₁ - a₁)i + (b₂ - a₂)j + (b₃ - a₃)k

Given Parameter

The displacement from the start to the finish is$\left(16i+12j\right)-\left(5i+3j\right)=\left(11i+9j\right)$

Step 4:Position vector of Point A

The displacement from the starting point to A is$\left(11i+9j\right)\times f$

The position vector of point A is$\left(5i+3j\right)+\left(11i+9j\right)\times f=\left(5+11f\right)i+\left(3+9f\right)j$

Position Vector When f=0

For f = 0 we have the position vector$\left(5+11\times 0\right)i+\left(3+9\times 0\right)j=5i+3j$

Check Reasonability

This is reasonable because it is the location of the starting point, 5i+3j

Step 7:Position Vector When f=1

For f = 1 we have position vector$\left(5+11\right)i+\left(3+9\right)j=16i+12j$

Check Reasonbility

This is reasonable because we have completed the trip and this is the position vector of the endpoint