Coincident, parallel, intersecting, skew lines practice questions.

The lines L1: r = (1, 2, 3) + λ(2, -1, 4) and L2: r = (5, 0, 11) + μ(4, -2, 8) are given. Determine the relationship between L1 and L2.

• They are parallel and distinct
• They are skew
• They are coincident
• They intersect at exactly one point

Answer: They are coincident

The direction vectors satisfy (4, -2, 8) = 2(2, -1, 4), so the lines are parallel; substituting λ = 2 into L1 gives (5, 0, 11), which lies on L2, so the lines are the same line (coincident).

Consider L1: r = (1, 0, 2) + λ(1, 2, -1) and L2: r = (0, 3, 1) + μ(2, 4, -2). Which statement correctly describes the two lines?

• They are coincident
• They are parallel and distinct
• They intersect at exactly one point
• They are skew

Answer: They are parallel and distinct

Since (2, 4, -2) = 2(1, 2, -1) the lines are parallel; testing (0, 3, 1) in L1 gives λ = -1 from the x-equation but then y = -2 ≠ 3, so the point is not on L1 and the lines are distinct.

The lines L1: r = (1, 2, 0) + λ(1, 0, 1) and L2: r = (0, 5, 1) + μ(0, 1, 1) have direction vectors that are not scalar multiples of each other. Solving the x- and y-equations gives λ = -1 and μ = -3, but these values do not satisfy the z-equation. State the relationship between the lines.

• They are skew
• They are parallel
• They intersect at exactly one point
• They are coincident

Answer: They are skew

Non-parallel direction vectors rule out parallel and coincident; since the system of equations is inconsistent the lines never meet, so they are skew.

The lines L1: r = (1, 2, 3) + λ(2, 1, 1) and L2: r = (0, k, 1) + μ(1, 1, 2) intersect. Determine the value of k.

• k = 2
• k = 1
• k = 0
• k = -1

Answer: k = 1

Solving the x- and z-equations gives λ = 0 and μ = 1; substituting into the y-equation gives 2 = k + 1, so k = 1.

Two lines in three-dimensional space have direction vectors that are NOT scalar multiples of each other. Which statement must be true?

• The lines intersect at exactly one point
• The lines are skew
• The lines are coincident
• They intersect once or are skew

Answer: They intersect once or are skew

Non-parallel direction vectors exclude parallel and coincident lines, but in three dimensions the lines may still fail to meet, so they are either intersecting or skew.

Which pair of direction vectors could belong to two parallel lines?

• (1, 2, 3) and (2, 3, 4)
• (1, -1, 2) and (2, -2, -4)
• (3, 0, 1) and (0, 3, 1)
• (2, -4, 6) and (-1, 2, -3)

Answer: (2, -4, 6) and (-1, 2, -3)

Parallel lines need direction vectors that are scalar multiples: (2, -4, 6) = -2(-1, 2, -3), while none of the other pairs are multiples of each other.