Question

Consider the two vectors A(with arrow above)=5i-3j and B(with arrow above)=-i-2j

1. Calculate A (with arrow above) + B (with arrow above)
2. Calculate A (with arrow above) - B (with arrow above)
3. Calculate absolute value of A (with arrow above) + B (with arrow above)
4. Calculate absolute value of A (with arrow above) - B (with arrow above)
5. Calculate the directions of A (with arrow above) + B (with arrow above) and A (with arrow above) - B (with arrow above).
(answers are counterclockwise from the +xaxis)

Answer 1

\[A+B=(5i-3j)+(-i-2j)=(5i-i)+(-3j-2j)=4i-5j\]\[A-B=(5i-3j)-(-i-2j)=(5i+i)+(-3j+2j)=6i-j\]

Answer 2

\[\left| A \right|-\left| B \right|=\sqrt{34}-\sqrt{5}\]\[\left| A \right|+\left| B \right|=\sqrt{5^{2}+(-3)^{2}}+\sqrt{(-1)^{2}+(-2)^{2}}=\sqrt{25+9}+\sqrt{1+4}=\sqrt{34}+\sqrt{5}\]

Answer 3

For directions, do you mean the components of the vector?

Answer 4

3. The absolute value thing means "magnitude" for vectors. Mathematicians call it the norm. It basically means the length of the vector.

So... the magnitude of A+B is

\[\left| A+B \right| = \left| 4i - 5j \right|= \sqrt{4^2 + 5^2} = \sqrt{41}\]

You just add them using the Pythagorean theorem.
Remember, the minus sign just means in the negative j direction. Usually "down"

4. Same as above.

5. Okay, I'm gonna teach you a trick. If your in college this will save your retriceon an exam and save you alot of time.

When you're asked to find the direction of a vector, and ONLY IF your coordinate system isn't something perverted, like twisted upsidedown or something, this will work.

\[\theta = arcTan(y/x)

Arctan is the tan^-1 thing usually on calculators. y is the vertical direction, j, and x is the horizontal direction, i. REMEMBER NEGATIVE SIGNS!!!! You're calc will give you a direction in degrees (if you tell it to) and that's your answer.