Question

three kids are pushing on a large earth ball in the middle of the playground. one child is pushing along the vector $-2\mathbf{i} - 16\mathbf{j}$, and the second child is pushing along the vector $-9\mathbf{i} + 7\mathbf{j}$. along what vector must the third child be pushing the ball if it is remaining still?

Explanation:

Step1: Define equilibrium condition

For the ball to stay still, the sum of all vectors must be $\vec{0}$. Let the third child's vector be $\vec{v} = a\vec{i} + b\vec{j}$.

Step2: Sum the given vectors

Add the first two vectors:
$$(-2\vec{i} - 16\vec{j}) + (-9\vec{i} + 7\vec{j}) = (-2-9)\vec{i} + (-16+7)\vec{j} = -11\vec{i} -9\vec{j}$$

Step3: Solve for the third vector

Set the total sum to $\vec{0}$:
$$-11\vec{i} -9\vec{j} + a\vec{i} + b\vec{j} = 0\vec{i} + 0\vec{j}$$
Equate components:
For $\vec{i}$: $-11 + a = 0 \implies a = 11$
For $\vec{j}$: $-9 + b = 0 \implies b = 9$
So $\vec{v} = 11\vec{i} + 9\vec{j}$

Answer:

$11\vec{i} + 9\vec{j}$