Finding parallel values for two-dimensional vectors

Key takeaway: This question uses the parallel-vector condition, which compares corresponding components through a constant scale factor.

Use the parallel vector condition

Two vectors are parallel when one is a scalar multiple of the other. For

$

$a=7\hat i+6\hat j\quad \text{and} \quad b=2\hat i+x\hat j$

$

the components must satisfy the same ratio:

$

$\frac{7}{2}=\frac{6}{x}$

$

Solve for x

Cross-multiply to get

$

$7x=12$

so

$

$x=\frac{12}{7}$

$

Why this works

If two vectors point in the same or exactly opposite direction, their components are proportional. That means every component changes by the same scale factor. Here, the scale factor from $b$ to $a$ is $\frac{7}{2}$ for the $i$-components, so the $j$-components must match that same ratio.

Final answer

The value of $x$ that makes $a$ parallel to $b$ is

$

$\boxed{\frac{12}{7}}$

$

---

Pitfalls the pros know 👇 A frequent error is to set the components equal, such as writing $7=2$ or $6=x$. Parallel vectors do not need equal components; they need proportional components. Another mistake is to forget that negative scalar multiples also count as parallel, so both same-direction and opposite-direction cases are included. In this specific problem, the ratio method directly gives the needed value of $x$.

What if the problem changes? If the question were changed to ask for vectors that are perpendicular instead of parallel, you would use the dot product condition $a\cdot b=0$ rather than a ratio. If the vector were $b=2\hat i-x\hat j$, then the parallel condition would become $\frac{7}{2}=\frac{6}{-x}$, giving a different value of $x$. The method changes with the relationship being tested, so always read whether the task asks for parallel, perpendicular, or equal vectors.

Tags: scalar multiple, vector ratio, parallel vectors