Find the vector equation of the line passing through A(2, −1, 1) and B(4, 1, 5). Check whether C(1, −2, −1) lies on this line.
How to lay out the answer. Follow the frame, then write it in your own words.
1 · Write what is given and what is to be found. Given: A(2, −1, 1), B(4, 1, 5) and C(1, −2, −1). To find: the vector equation of AB, and whether C is on it.
2 · Draw the figure or sketch. Draw a line, mark A and B on it, and mark the point C to be tested.
3 · Write the formula. r⃗ = a⃗ + λ(b⃗ − a⃗), where a⃗ and b⃗ are the position vectors of A and B.
4 · Find the equation.
a⃗ = 2î − ĵ + k̂, b⃗ = 4î + ĵ + 5k̂
Direction: b⃗ − a⃗ = 2î + 2ĵ + 4k̂
r⃗ = (2î − ĵ + k̂) + λ(2î + 2ĵ + 4k̂)
5 · Test the point C.
General point: (2 + 2λ, −1 + 2λ, 1 + 4λ)
2 + 2λ = 1 gives λ = −1/2
y: −1 + 2(−1/2) = −2, which matches
z: 1 + 4(−1/2) = −1, which matches
6 · Check and box the answer. r⃗ = (2î − ĵ + k̂) + λ(2î + 2ĵ + 4k̂). C lies on the line, at λ = −1/2.
Scoring tip: When you test a point, show all three components. One matching component proves nothing.
