Question

sketch the vector as a position vector and find its magnitude.
\\( \mathbf{v} = -7 \mathbf{i} - 6 \mathbf{j}

choose the correct answer below.
\\( \bigcirc \\) a.
\\( \bigcirc \\) b.
\\( \bigcirc \\) c.
\\( \bigcirc \\) d.

Explanation:

Step1: Identify vector components

The vector $\mathbf{v} = -7\mathbf{i} -6\mathbf{j}$ has $x$-component $v_x = -7$ and $y$-component $v_y = -6$. A position vector starts at the origin $(0,0)$ and ends at $(v_x, v_y) = (-7, -6)$, so the arrow points from $(0,0)$ to $(-7,-6)$. This matches option A.

Step2: Calculate vector magnitude

Use the magnitude formula for a 2D vector $\mathbf{v} = a\mathbf{i} + b\mathbf{j}$: $|\mathbf{v}| = \sqrt{a^2 + b^2}$
Substitute $a=-7$, $b=-6$:
$$|\mathbf{v}| = \sqrt{(-7)^2 + (-6)^2}$$

Step3: Simplify the expression

Calculate the squares and sum:
$$|\mathbf{v}| = \sqrt{49 + 36} = \sqrt{85}$$

Answer:

Correct graph: A.
Magnitude: $\sqrt{85}$