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Math  /  Geometry

Question```latex Determine whether the lines r⃗(λ)\vec{r}(\lambda) and s⃗(μ)\vec{s}(\mu) intersect, and, if they do intersect, find the point of intersection. ```

Studdy Solution

STEP 1

1. The lines r⃗(λ)\vec{r}(\lambda) and s⃗(μ)\vec{s}(\mu) are given in parametric vector form.
2. The parameters λ\lambda and μ\mu are real numbers.
3. The lines intersect if there exists a common point P⃗\vec{P} such that r⃗(λ)=s⃗(μ)\vec{r}(\lambda) = \vec{s}(\mu).

STEP 2

1. Write the parametric equations for both lines.
2. Set the parametric equations equal to each other to find the values of λ\lambda and μ\mu.
3. Verify if the obtained values of λ\lambda and μ\mu satisfy the equations.
4. Determine the point of intersection, if it exists.

STEP 3

Write the parametric equations for both lines:
Assume the lines are given by: r⃗(λ)=a⃗+λb⃗\vec{r}(\lambda) = \vec{a} + \lambda \vec{b} s⃗(μ)=c⃗+μd⃗\vec{s}(\mu) = \vec{c} + \mu \vec{d}
Where a⃗\vec{a}, b⃗\vec{b}, c⃗\vec{c}, and d⃗\vec{d} are vectors in R3\mathbb{R}^3.

STEP 4

Set the parametric equations equal to each other to find the values of λ\lambda and μ\mu:
a⃗+λb⃗=c⃗+μd⃗\vec{a} + \lambda \vec{b} = \vec{c} + \mu \vec{d}
This results in a system of linear equations. Solve for λ\lambda and μ\mu.
STEP_2.1: Solve the system of equations obtained from equating the components of the vectors:
1. a1+λb1=c1+μd1 a_1 + \lambda b_1 = c_1 + \mu d_1
2. a2+λb2=c2+μd2 a_2 + \lambda b_2 = c_2 + \mu d_2
3. a3+λb3=c3+μd3 a_3 + \lambda b_3 = c_3 + \mu d_3

STEP 5

Verify if the obtained values of λ\lambda and μ\mu satisfy all three equations:
Substitute λ\lambda and μ\mu back into the equations to check for consistency.

STEP 6

Determine the point of intersection, if it exists:
If the values of λ\lambda and μ\mu satisfy all equations, substitute λ\lambda into r⃗(λ)\vec{r}(\lambda) or μ\mu into s⃗(μ)\vec{s}(\mu) to find the intersection point P⃗\vec{P}.
If the lines intersect, the point of intersection is:
P⃗=r⃗(λ)=s⃗(μ) \vec{P} = \vec{r}(\lambda) = \vec{s}(\mu)
If no consistent solution exists, the lines do not intersect.

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