Back to the full dot-point answer

NSWMaths Extension 1Quick questions

Vectors (ME-V1)

Quick questions on Parametric vector equations of lines: point and direction form, parameter elimination

14short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is vector equation of a line?
Show answer
A line through point AA with direction vector d\mathbf{d} (non-zero) is the set of points
What is line through two points?
Show answer
A line through points AA and BB has direction vector AB=b−a\mathbf{AB} = \mathbf{b} - \mathbf{a}. So
What is converting to Cartesian form?
Show answer
From the parametric form, eliminate λ\lambda.
What is vertical and horizontal lines?
Show answer
If d1=0d_1 = 0, the line is vertical: x=a1x = a_1 for all λ\lambda.
What is intersection of two lines?
Show answer
Set the parametric equations of two lines equal:
What is two interpretations of λ\lambda?
Show answer
The latter is useful for collision problems: do two particles meet, and if so when?
What is convert to Cartesian?
Show answer
Line r=(1,3)+λ(2,5)\mathbf{r} = (1, 3) + \lambda (2, 5).
What is intersection?
Show answer
Lines L1:r=(1,0)+λ(1,2)L_1: \mathbf{r} = (1, 0) + \lambda (1, 2) and L2:r=(0,4)+μ(1,−1)L_2: \mathbf{r} = (0, 4) + \mu (1, -1). Find the intersection.
What is collision problem?
Show answer
Two particles. Particle 1 starts at (0,0)(0, 0) with velocity (2,1)(2, 1). Particle 2 starts at (8,3)(8, 3) with velocity (−1,1)(-1, 1). Do they collide, and if so when?
What is parallel lines?
Show answer
Lines L1:r=(1,1)+λ(2,3)L_1: \mathbf{r} = (1, 1) + \lambda (2, 3) and L2:r=(4,0)+μ(4,6)L_2: \mathbf{r} = (4, 0) + \mu (4, 6).
What is direction vector scaling?
Show answer
(2,6)(2, 6) and (1,3)(1, 3) are the same direction. The vector equation can use either; the parameter λ\lambda runs over different values.
What is parallel vs intersecting?
Show answer
Two lines are parallel iff their direction vectors are scalar multiples of each other; otherwise they intersect.
What is confusing line equation with point on line?
Show answer
r=(1,2)+λ(3,−1)\mathbf{r} = (1, 2) + \lambda (3, -1) is the equation. The specific point at λ=1\lambda = 1 is (4,1)(4, 1).
What is collision vs path-crossing?
Show answer
Two particles whose paths cross do not necessarily collide; collision requires being at the same place at the same time. :::

All Maths Extension 1Q&A pages