Explained: "How do you find the vector parametrization of..."

runddansm88e

Answered question

2022-04-11

How do you find the vector parametrization of the line of intersection of two planes

2 x - y - z = 5 and x - y + 3 z = 2

cloirdxti

Beginner2022-04-12Added 12 answers

Step 1

x = t

z = \frac{1}{4} t - \frac{3}{4}

y = \frac{7}{4} t - \frac{17}{4}

Let's solve the system of the two equations, explaining two letters in function of the third:

2 x - y - z = 5

x - y + 3 z = 2

y = 2 x - z - 5

x - (2 x - z - 5) + 3 z = 2 \Rightarrow x - 2 x + z + 5 + 3 z = 2 \Rightarrow

4 z = x - 3 \Rightarrow z = \frac{1}{4} x - \frac{3}{4}

Step 2
Therefore,

y = 2 x - (\frac{1}{4} x - \frac{3}{4}) - 5 \Rightarrow y = 2 x - \frac{1}{4} x + \frac{3}{4} - 5

y = \frac{7}{4} x - \frac{17}{4}

Thus,

z = \frac{1}{4} x - \frac{3}{4}

y = \frac{7}{4} x - \frac{17}{4}

Step 3
and if we put

x = t

x = t

z = \frac{1}{4} t - \frac{3}{4}

y = \frac{7}{4} t - \frac{17}{4}

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