Math worksheet for practicing vector addition and subtraction using the graphical method.

To solve these vector problems using the graphical method, we use two main rules:

1. Vector Addition ($A + B$): Place the tail of the second vector at the head (tip) of the first vector. The result (resultant) is a new arrow drawn from the start of the first vector to the end of the second vector. This is often called the "Tip-to-Tail" method.
2. Vector Subtraction ($A - B$): This is the same as adding a negative vector ($A + (-B)$). To get $-B$, you flip vector $B$ so it points in the exact opposite direction. Then, add it to $A$ using the Tip-to-Tail method. Alternatively, for $A - B$, you can draw an arrow from the tip of $B$ to the tip of $A$ if they both start from the same point.

Let's break down each problem step-by-step based on the vectors shown in the grid. Note that in the diagrams, Vector $A$ generally points Up and Right, and Vector $B$ generally points Down and Right.

Problem 1: Draw $A + B$
* Start with Vector $A$.
* Take Vector $B$ and move it so its tail touches the tip of $A$. Keep $B$ pointing in the same direction (down and right).
* Draw the resultant vector from the starting point (tail of $A$) to the final endpoint (tip of the moved $B$).
* *Result:* A long vector pointing mostly to the right, slightly up or down depending on lengths, but visually it connects the start of $A$ to the end of $B$.

Problem 2: Draw $A - B$
* This means $A + (-B)$.
* First, reverse Vector $B$ to make $-B$. If $B$ pointed down-right, $-B$ points up-left.
* Place the tail of $-B$ at the tip of $A$.
* Draw the resultant from the tail of $A$ to the tip of $-B$.
* *Alternative Visual Check:* If you look at the original positions, draw an arrow from the tip of $B$ to the tip of $A$. That represents $A - B$.

Problem 3: Draw $B - A$
* This means $B + (-A)$.
* Reverse Vector $A$ to make $-A$ (points down-left).
* Place the tail of $-A$ at the tip of $B$.
* Draw the resultant from the tail of $B$ to the tip of $-A$.
* *Alternative Visual Check:* Draw an arrow from the tip of $A$ to the tip of $B$. That represents $B - A$. Note that this is the exact opposite of Problem 2.

Problem 4: Draw $B + A$
* Start with Vector $B$.
* Place the tail of $A$ at the tip of $B$.
* Draw the resultant from the tail of $B$ to the tip of $A$.
* *Note:* In vector math, $A + B$ is the same as $B + A$. The answer should look identical to Problem 1.

Problem 5: Draw $B - A$
* This is the same instruction as Problem 3.
* Draw the vector from the tip of $A$ to the tip of $B$.

Problem 6: Draw $B + A$
* This is the same instruction as Problem 4 (and effectively Problem 1).
* Connect tail of first to tip of second. Result is the same diagonal as #1 and #4.

Problem 7: Draw $A - B$
* Same as Problem 2.
* Draw the vector from the tip of $B$ to the tip of $A$.

Problem 8: Draw $A - B$
* Same as Problem 2 and 7.
* Draw the vector from the tip of $B$ to the tip of $A$.

Problem 9: Draw $B - A$
* Same as Problem 3 and 5.
* Draw the vector from the tip of $A$ to the tip of $B$.

Problem 10: Draw $B - A$
* Same as Problem 3, 5, and 9.
* Draw the vector from the tip of $A$ to the tip of $B$.

Problem 11: Draw $B + A$
* Same as Problem 1, 4, and 6.
* Add them tip-to-tail. The resultant goes from the very beginning to the very end.

Problem 12: Draw $B + A$
* Same as Problem 1, 4, 6, and 11.
* Add them tip-to-tail.

Summary of How to Draw Each:

* For any "$+$" problem ($A+B$ or $B+A$):
1. Slide one vector so its tail touches the other vector's tip.
2. Draw a new arrow from the empty tail to the empty tip.
3. The resulting arrow will be longer and point diagonally across the space covered by both.

* For any "$-$" problem where it is $A - B$:
1. Look at the tips of the original vectors $A$ and $B$.
2. Draw an arrow starting at the tip of $B$ and ending at the tip of $A$.

* For any "$-$" problem where it is $B - A$:
1. Look at the tips of the original vectors $A$ and $B$.
2. Draw an arrow starting at the tip of $A$ and ending at the tip of $B$.

Final Answer:
The solutions require drawing arrows on the provided grids. Here is the description of the correct drawing for each number:

1. $A + B$: Draw a vector starting at the tail of $A$ and ending at the tip of $B$ (after moving $B$'s tail to $A$'s tip).
2. $A - B$: Draw a vector connecting the tip of $B$ to the tip of $A$.
3. $B - A$: Draw a vector connecting the tip of $A$ to the tip of $B$.
4. $B + A$: Same result as #1. Draw the resultant diagonal from start to finish.
5. $B - A$: Same result as #3. Draw vector from tip of $A$ to tip of $B$.
6. $B + A$: Same result as #1. Draw the resultant diagonal.
7. $A - B$: Same result as #2. Draw vector from tip of $B$ to tip of $A$.
8. $A - B$: Same result as #2. Draw vector from tip of $B$ to tip of $A$.
9. $B - A$: Same result as #3. Draw vector from tip of $A$ to tip of $B$.
10. $B - A$: Same result as #3. Draw vector from tip of $A$ to tip of $B$.
11. $B + A$: Same result as #1. Draw the resultant diagonal.
12. $B + A$: Same result as #1. Draw the resultant diagonal.