Question

sketch the vector as a position vector and find its magnitude.
( mathbf{v} = 4mathbf{i} + mathbf{j} )

choose the correct graph below.

\\( \bigcirc \\) a.

\\( \bigcirc \\) b.

\\( \bigcirc \\) c.

\\( \bigcirc \\) d.

Explanation:

Step1: Identify vector components

The vector $\mathbf{v}=4\mathbf{i}+\mathbf{j}$ has $x$-component $4$ and $y$-component $1$, so it terminates at $(4,1)$. This matches graph C, as it starts at the origin and ends at the point in the first quadrant with positive $x$ and small positive $y$.

Step2: Calculate vector magnitude

Use the magnitude formula for $\mathbf{v}=a\mathbf{i}+b\mathbf{j}$: $\|\mathbf{v}\|=\sqrt{a^2+b^2}$
$$\|\mathbf{v}\|=\sqrt{4^2+1^2}$$

Step3: Simplify the expression

$$\|\mathbf{v}\|=\sqrt{16+1}=\sqrt{17}$$

Answer:

Correct graph: C.
Magnitude of the vector: $\sqrt{17}$