Question 1207395: The lines l and m have vector equations
r = 2i-j+4k+s(i+j- k) and r= -2i + 2j+ k+ t(-2i+j+ k)
respectively・
i) Show that l and m do not intersect.
The point P lies on / and the point @ has position vector 2i - k.
ii)Given that the line PQ is perpendicular to l, find the position vector of P.
(iii) Verify that Q lies on m and that PQ is perpendicular to m.
Answer by ElectricPavlov(122)   (Show Source):
You can put this solution on YOUR website! **i) Show that l and m do not intersect**
* **Find the direction vectors of the lines:**
* Direction vector of line l: **d₁** = i + j - k
* Direction vector of line m: **d₂** = -2i + j + k
* **Check if the lines are parallel:**
* Lines are parallel if their direction vectors are scalar multiples of each other.
* **d₁** is not a scalar multiple of **d₂**, so the lines are not parallel.
* **Check for intersection:**
* To find the point of intersection, we need to find values of 's' and 't' that satisfy the following equations:
* 2 + s = -2 - 2t
* -1 + s = 2 + t
* 4 - s = 1 + t
* Solving this system of equations, we find that there is no consistent solution for 's' and 't'.
* **Conclusion:** Since the lines are not parallel and do not intersect, they are **skew lines**.
**ii) Find the position vector of P**
* **Let P be the point on line l with position vector:**
* **r_P = 2i - j + 4k + s(i + j - k)**
* **Find the direction vector of PQ:**
* **PQ = OQ - OP = (2i - k) - (2i - j + 4k + s(i + j - k))**
* **PQ = -j - 5k - s(i + j - k)**
* **Since PQ is perpendicular to l, their dot product must be zero:**
* **PQ • d₁ = 0**
* (-j - 5k - s(i + j - k)) • (i + j - k) = 0
* -s - j + k - s(i + j - k) = 0
* -s - j + k - s - s = 0
* -3s - j + k = 0
* **Solve for 's':**
* This equation implies that s = 0.
* **Find the position vector of P:**
* **r_P = 2i - j + 4k + 0(i + j - k)**
* **r_P = 2i - j + 4k**
**iii) Verify that Q lies on m and that PQ is perpendicular to m**
* **Check if Q lies on line m:**
* The position vector of Q is 2i - k.
* We need to find a value of 't' such that:
* 2i - k = -2i + 2j + k + t(-2i + j + k)
* 4i - 2j - 2k = t(-2i + j + k)
* Comparing coefficients, we can see that t = -2 satisfies the equation.
* **Check if PQ is perpendicular to m:**
* **PQ = -j - 5k** (from part ii)
* **d₂ = -2i + j + k** (direction vector of line m)
* **PQ • d₂ = (-j - 5k) • (-2i + j + k) = 0 - 1 - 5 = -6**
* Since the dot product of PQ and d₂ is not zero, PQ is **not** perpendicular to m.
**Conclusion:**
* There is an error in the problem statement or the calculations.
* If Q lies on line m, then PQ should be perpendicular to m.
**Please double-check the given information and the calculations in part ii) to ensure accuracy.**
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