Let i and j be unit vectors of the coordinate axes.…

Let $\vec{i}$ and $\vec{j}$ be unit vectors of the coordinate axes. Construct vectors $\vec{a} = 4\vec{i}+\vec{j}$ and $\vec{b} = \vec{i}+3\vec{j}$. What are the projections of $\vec{a}$ and $\vec{b}$ onto the coordinate axes?

Step 1: Component form

The vectors are given directly in terms of the unit vectors $\vec{i}$ (along the $x$-axis) and $\vec{j}$ (along the $y$-axis): \[\vec{a} = 4\vec{i} + \vec{j} \;\Longrightarrow\; \vec{a} = (4,\,1), \qquad \vec{b} = \vec{i} + 3\vec{j} \;\Longrightarrow\; \vec{b} = (1,\,3).\]

Step 2: Projections onto the coordinate axes

For $\vec{v} = v_x\vec{i} + v_y\vec{j}$, the projection onto the $x$-axis is $v_x$ and onto the $y$-axis is $v_y$.

Projections of $\vec{a} = 4\vec{i}+\vec{j}$

\[a_x = 4, \qquad a_y = 1\]

Projections of $\vec{b} = \vec{i}+3\vec{j}$

\[b_x = 1, \qquad b_y = 3\]

Step 3: Magnitudes and angles

For $\vec{a}$

\[ |\vec{a}| = \sqrt{4^2+1^2} = \sqrt{17}, \qquad \tan\alpha_a = \frac{a_y}{a_x} = \frac{1}{4} \;\Longrightarrow\; \alpha_a = \arctan\frac{1}{4} \approx 14.04^\circ \]

For $\vec{b}$

\[ |\vec{b}| = \sqrt{1^2+3^2} = \sqrt{10}, \qquad \tan\alpha_b = \frac{b_y}{b_x} = 3 \;\Longrightarrow\; \alpha_b = \arctan 3 \approx 71.57^\circ \]

Step 4: Graphical construction