Compare coefficients in two independent directions

Compare coefficients in two independent directions

Write the unknown point in two ways using the same origin and the same two independent directions. Equate the coefficients of each direction, solve the scalar equations, then check what the parameters say about the point order.

Two nonzero, nonparallel vectors $\mathbf a,\mathbf b$ span their plane: any vector in that plane has form $\lambda\mathbf a+\mu\mathbf b$. Its coefficients are unique:

$$\lambda_1\mathbf a+\mu_1\mathbf b=\lambda_2\mathbf a+\mu_2\mathbf b\quad\Longleftrightarrow\quad\lambda_1=\lambda_2\text{ and }\mu_1=\mu_2.$$

Example 5’s answers show this form: $(7/10)\mathbf a+(3/10)\mathbf b$, $(7/5)\mathbf a-(2/5)\mathbf b$, and $-(3/4)\mathbf a+(7/4)\mathbf b$. They are combinations in the same plane, including negative coefficients for external points. Each pair sums to one because these particular points lie on line $AB$; arbitrary planar combinations need not have that sum.

Why independence matters

The nonparallel condition is essential. Subtract the two representations. If the difference of the $\mathbf b$ coefficients were nonzero, the resulting equation would make $\mathbf b$ a multiple of $\mathbf a$, a contradiction. Both coefficient differences must therefore vanish. Parallel directions could give different coefficient pairs for the same vector.

Use two representations of the intersection

Example 7 uses $P$ midpoint of $AB$, $Q$ midpoint of $OP$ and $R=AQ\cap OB$. Describe $R$ from each line, keeping a common origin:

$$\overrightarrow{OR}=k\mathbf b=\left(1-\frac{3\lambda}4\right)\mathbf a+\frac\lambda4\mathbf b.$$

Compare coefficients only after identifying the independent directions. The source calculation gives $\lambda=4/3$, $k=1/3$; $\lambda>1$ confirms $R$ lies on $AQ$ produced. The full exercise keeps both midpoint steps, the position-to-displacement subtraction, and the final ratio $AQ:QR=3:1$.