Finding Vector Paths Flashcards

All 6 cards in this deck

If $\vec{OA}=\mathbf{a}$ and $\vec{OB}=\mathbf{b}$, what is $\vec{AB}$ in terms of $\mathbf{a}$ and $\mathbf{b}$?

$\mathbf{b}-\mathbf{a}$

(Go backwards along $\mathbf{a}$ to $O$, then forwards along $\mathbf{b}$.)

How is $\vec{BA}$ related to $\vec{AB}$?

$\vec{BA}=-\vec{AB}$ — same length, opposite direction.

What are the steps to find a vector path in terms of $\mathbf{a}$ and $\mathbf{b}$?

e.g. find $\vec{AD}$ where $\vec{OA}=\mathbf{a}$ and $\vec{OD}=-\mathbf{b}$

1. Choose a route along known vectors: $A\to O\to D$
2. Write each step, using a minus sign when travelling backwards along a vector: $-\mathbf{a}$ then $-\mathbf{b}$
3. Add the steps: $\vec{AD}=-\mathbf{a}-\mathbf{b}$

$M$ is the midpoint of $AB$ and $\vec{AB}=\mathbf{b}$. What is $\vec{AM}$?

$\frac{1}{2}\mathbf{b}$

True or false? If $M$ is the midpoint of $AB$ and $\vec{AB}=2\mathbf{a}$, then $\vec{MB}=\mathbf{a}$.

True. $\vec{AM}$ and $\vec{MB}$ are each half of $\vec{AB}$.

How can you tell that two vectors written in terms of $\mathbf{a}$ and $\mathbf{b}$ are parallel?

e.g. $\mathbf{a}+\mathbf{b}$ and $2\mathbf{a}+2\mathbf{b}$

One can be written as a scalar multiple of the other.

e.g. $2\mathbf{a}+2\mathbf{b}=2(\mathbf{a}+\mathbf{b})$, so they are parallel.