Solving Vector Applications: a-b Magnitude & Ball's Height

[Huskies213]

Can anyone explain what to do for these 2 ?

1.) If a=3i-4j, and b=4i-3j, what is the magnitude of the vector a-b?


2.) A boy throws a ball at an initial velocity of 26m/s at an angle of 20 degrees above the horizontal. How high above the projection point is the ball after 1.4 seconds ?

 

[Astronuc]

For the vector problem, i,j represent usually represent mutually orthogonal (perpendicular) vectors, e.g. in x,y-directions.

To add two vectors, one adds corresponding components.

let a = a₁i + a₂j, and b = b₁i + b₂j, so a + b = (a₁+b₁)i + (a₂+b₂)j. The subtraction is just the additive inverse, i.e. replace + with -.

The magnitude of a is just sqrt(a₁²+a₂²). Remember the formula for the length of the hypotenuse of a right triangle - Pythagorean theorem.

In question 2, one has to resolve the vector given by "26m/s at an angle of 20 degrees" into horizontal and vertical components. The ball travels vertically with some velocity component, but decelerates due to gravity, reaches a maximum at some time T, and returns to the same initial elevation at time 2T (i.e. T up and T down - neglecting air resistance).

Using the vertical velocity component, one can establish the equation of motion as a function of time to determine where the ball is at 1.4 s. If T (time to max height) < 1.4 s, then the ball is falling back from its maximum altitude.