9.2 Vectors in the Plane

Vectors in the Plane

Let P=(1,3) and Q=(2,1) be points in (ℝ^2).

Graph these points.

Graph the vector (OverVector[PQ]).  What are its components?  
(Subscript[((OverVector[PQ])), x])≡c=____      (Subscript[((OverVector[PQ])), y])≡d=____

Graph the equivalent vector with base at the origin.

Graph an equivalent vector that does not have its base at the origin.

Graph a parallel vector that is not equivalent.

Determine the length of the vector (OverVector[PQ]).  Explain why this is the length.

Draw the line that passes through the points P and Q.

Find the equation for this line in the form y=m x+b.

Express m in terms of the components c and d of (OverVector[PQ]).

Find a parametric equation, in the form {( x(t),y(t) )=(a,b)+t (e,f):0<t<1} , describing the coordinates of the points on the line segment between P and Q. [Hint to get you started: when t=0, you only have to worry about finding a and b.]

Consider the graphed vectors.

Find the components of the vectors u= (OverVector[OP]) and v=(OverVector[OQ]).   

u=<_____, _____>

v=<_____, _____>

Determine 3u and u+v numerically and graphically.

Write w=⟨2,8⟩ as a linear combination of u and v.  (A “linear combination” means adding and/or subtracting scalar multiples of of u and v.)