Vector Examples, Equations, and Practice Problems

Find the vector that has the same direction as (6, 2, -3) but is four lengths.

1)What is the position vector r(t) as a function of angle ${\displaystyle \theta \left(t\right)}$. For later remember that ${\displaystyle \theta \left(t\right)}$ is itself a function of time.
Give your answer in terms of ${\displaystyle R,\theta \left(t\right)}$, and unit vectors x and y corresponding to the coordinate system in thefigure. 2)For uniform circular motion, find ${\displaystyle \theta \left(t\right)}$ at an arbitrary time t.
Give your answer in terms of ${\displaystyle \omega }$ and t.
3)Find r, a position vector at time.
Give your answer in terms of R and unit vectors x and/or y.
4)Determine an expression for the positionvector of a particle that starts on the positive y axis at (i.e., at ,$(x_0,y_0)=(0,R)$) and subsequently moves with constant ${\displaystyle \omega }$.
Express your answer in terms of R, $\omega$ ,t ,and unit vectors x and

This is the quesetion. Suppose that a does not equal 0.
a. if ${\displaystyle a\cdot b=a\cdot c}$, does it follow that $b=c$?
b. if ${\displaystyle a\times b=a\times c}$, does it follow that $b=c$ ?
c. if ${\displaystyle a\cdot b=a\cdot c}$ and ${\displaystyle a\times b=a\times c}$, does it follow that $b=c$?
Either prove the assertion is true in general or show that it is false for a concret choice of vectors a, b, c