Drawing lines and planes from parametric and implicit sets

Key concept: This problem tests how to interpret subset descriptions of $\mathbb{R}^2$ and $\mathbb{R}^3$ in geometric form, then sketch the resulting line or plane correctly.

Step by step

Identify the geometric meaning of each set

Each description is either an implicit equation or a parametric form. The key skill is translating algebra into geometry: a single linear equation in $\mathbb{R}^2$ usually gives a line, a single linear equation in $\mathbb{R}^3$ gives a plane, and a vector equation with one scalar parameter gives a line through a point in the direction of a vector.

For (i), the set

$\{\mathbf{x}=(x_1,x_2)\in\mathbb{R}^2\mid x_1=1\}$

is the vertical line where every point has first coordinate 1. So the graph is a line parallel to the $x_2$-axis.

For (ii),

$\mathbf{x}=a(2,1),\ a\in\mathbb{R},$

is a line through the origin in direction $(2,1)$. Every point on the set is a scalar multiple of $(2,1)$.

Use direction vectors and point-direction form

For (iii),

$\mathbf{x}=a(1,1)+(0,-1),\ a\in\mathbb{R},$

the line passes through the point $(0,-1)$ and has direction vector $(1,1)$. This is the point-direction form of a line. A convenient coordinate form is

$x_1=a,\qquad x_2=a-1,$

so the line has slope 1 and y-intercept $-1$.

For (iv),

$\{\mathbf{x}=(x_1,x_2,x_3)\in\mathbb{R}^3\mid x_1=x_2=x_3\},$

the set consists of all points on the diagonal line through the origin with direction $(1,1,1)$. It is a one-dimensional line in three-dimensional space.

Distinguish lines from planes in three dimensions

For (v),

$\mathbf{x}=a(1,0,0)+b(0,1,0),\ a,b\in\mathbb{R},$

the vectors span the $x_1x_2$-plane. Since any point has the form $(a,b,0)$, the set is exactly the plane $x_3=0$.

Final sketches in words

• (i) vertical line $x_1=1$
• (ii) line through the origin with direction $(2,1)$
• (iii) line through $(0,-1)$ with direction $(1,1)$
• (iv) diagonal line $x_1=x_2=x_3$
• (v) coordinate plane $x_3=0$

A strong way to sketch these is to mark one point and one direction vector for each line, then use the span description to recognize whether the set is a line or a plane.

Pitfall alert

Students often confuse a line defined by one direction vector with a plane defined by two independent direction vectors. In part (v), the two vectors $(1,0,0)$ and $(0,1,0)$ are independent, so their span is a plane, not a line. Another common mistake is treating $x_1=x_2=x_3$ as three separate constraints that somehow define a point. In fact, those equalities leave one free parameter, so the set is a line. When converting parametric forms to sketches, always count how many free parameters remain.

Try different conditions

If part (ii) were changed to $\mathbf{x}=a(2,1)+(1,-3)$, the set would still be a line, but it would no longer pass through the origin. Its new reference point would be $(1,-3)$ and its direction would still be $(2,1)$. If part (v) were changed to $\mathbf{x}=a(1,0,0)+b(0,1,0)+c(0,0,1)$, then the set would become all of $\mathbb{R}^3$, because the three standard basis vectors span the entire space. These variants show that the number of independent direction vectors determines dimension.

Further reading

parametric form, span of vectors, coordinate plane