Vector Arithmetic

The correct answer is A. $\boldsymbol{a}+\boldsymbol{b}=\langle a_x + b_x, a_y + b_y \rangle$. This is the formula for adding two vectors in component form given in the section titled "Addition and Subtraction."

The correct answer is B. The magnitude is $\sqrt{5}$ and the direction is $\boldsymbol{u}=\langle \frac{1}{\sqrt{5}}, \frac{2}{\sqrt{5}} \rangle$. See the section "Magnitude and Direction" for their respective formulas.

The correct answer is C.$2\sqrt{3}$. After adding the two vectors, you can find the magnitude of the resulting vector using the magnitude formula found in the "Magnitude and Direction" section.

Add the two vectors: $$\begin{align*} \boldsymbol{u}+\boldsymbol{v} = \langle u_1+v_1,u_2+v_2 \rangle \\ \langle 3,2 \rangle + \langle -1,4 \rangle = \langle 2,2 \rangle \\ \end{align*}$$ Compute the magnitude: $$\begin{align*} ||\boldsymbol{u}||= \sqrt{(u_1)^2+(u_2)^2} \\ ||\boldsymbol{a}+\boldsymbol{b}|| = ||\langle 2,2 \rangle || \\ = \sqrt{2^2+2^2} = \sqrt{8} = 2 \sqrt{2} \end{align*}$$

The correct answer is D. The magnitude of the sum can be equal to or less than, but not greater than, the sum of the magnitudes of the individual vectors, depending on their directions. The two sums will be equal when the vectors point in the same direction. Otherwise, the magnitude of the sum will be shorter due to the triangle inequality theorem. To see a visual of this, go to the "Magnitude and Direction" section of this page.