Question

Help with question 1 and 2

Answer 1

#1
If <-3a,4kb> is parallel to <7a+b>,
then the coefficients of corresponding components are proportional, meaning
-3/7 = 4k/1.

Solve for k to get k=-3/(28).

#2
It would be a similar problem to the previous.  We can equate the given vector to the direction vector,<1/2,p>=<-3q,2>To find p and q, we equate coefficients in each component direction,1/2=-3q  => q=-1/6p=2resulting in <1/2, 2>.

Next step is to reduce the vector to a unit vector.We know that for any given vector <a,b>, its unit vector is (a/sqrt(a^2+b^2), b/sqrt(a^2+b^2)) [similar to pythagoras).

First calculate sqrt((1/2)^2+2^2)=sqrt(1/4+4)sqrt(17/4)=sqrt(17)/2
Unit vector required,<1/2,2>/(sqrt(17)/2)=<1/2,2>*2/sqrt(17)>=<1/sqrt(17),4/sqrt(17)>
Check: magnitude of vector = (1/sqrt(17))^2+(4/sqrt(17))^2=1/17+16/17=1  ok