Equations of Lines in 3D (Edexcel A-Level Further Mathematics): Quizzes

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Equations of Lines in 3D
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In the vector form r=a+λb\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}, what does r\mathbf{r} represent?

Position vector of any point on the line

In r=a+λb\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}, what does a\mathbf{a} represent?

Position vector of a fixed point

In r=a+λb\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}, what does b\mathbf{b} represent?

Direction vector of the line

What type of quantity is λ\lambda in the equation r=a+λb\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}?

A scalar parameter

How are the vector and Cartesian forms of a line in 3D related?

They are equivalent

Find the vector equation of a line through (2,−1,3)(2, -1, 3) with direction (1,0,2)(1, 0, 2).

r=(2,−1,3)+λ(1,0,2)\mathbf{r} = (2, -1, 3) + \lambda(1, 0, 2)

Convert r=(1,2,3)+λ(2,−1,4)\mathbf{r} = (1, 2, 3) + \lambda(2, -1, 4) to Cartesian form.

x−12=y−2−1=z−34\frac{x-1}{2} = \frac{y-2}{-1} = \frac{z-3}{4}

How is Cartesian form obtained from vector form?

By eliminating λ\lambda

In x−23=y+1−2=z−45\frac{x-2}{3} = \frac{y+1}{-2} = \frac{z-4}{5}, what is the direction vector?

(3,−2,5)(3, -2, 5)

Which is a common mistake when working with line equations?

Confusing position and direction vectors

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