Let L be the line passing through the point P=(−4, −3, −5) with direction vector →d=[3, −1, 3]T. Find the shortest distance d from the point P0=(1, −1, −1) to L, and the point Q on L that is closest to P0. Use the square root where needed to give an exact value for your answer.
d=(?)
Q=(?,?,?)
To find the shortest distance from a point to a line, we can use the formula:
d = |(P0 - P) cross →d| / |→d|
First, let's calculate P0 - P:
P0 - P = (1 - (-4), -1 - (-3), -1 - (-5)) = (5, 2, 4)
Next, let's calculate the cross product of (P0 - P) and →d:
(P0 - P) cross →d = (5, 2, 4) cross (3, -1, 3)
To find the cross product, we can use the determinant of the following matrix:
| i j k |
| 5 2 4 |
| 3 -1 3 |
Cross product = [(2 * 3 - 4 * -1), (5 * 3 - 4 * 3), (5 * -1 - 2 * 3)]
= [10, 3, -13]
Now, let's calculate the magnitude (length) of →d:
|→d| = sqrt(3^2 + (-1)^2 + 3^2)
= sqrt(9 + 1 + 9)
= sqrt(19)
Therefore, the shortest distance d from point P0 to line L is:
d = |(P0 - P) cross →d| / |→d|
= |(10, 3, -13)| / sqrt(19)
= sqrt(10^2 + 3^2 + (-13)^2) / sqrt(19)
= sqrt(100 + 9 + 169) / sqrt(19)
= sqrt(278) / sqrt(19)
To get the exact value, we'll keep the square roots separate:
d = sqrt(278) / sqrt(19)
Now, let's find the point Q on line L that is closest to P0.
We can use the equation of a line to find Q. The equation of a line passing through the point P and with direction vector →d can be written as:
Q = P + t * →d
To find t, we can use the dot product:
(P0 - P) · →d = |P0 - P| * |→d| * cosθ
Where cosθ is the cosine of the angle between (P0 - P) and →d.
Let's calculate (P0 - P) · →d:
(P0 - P) · →d = (5, 2, 4) · (3, -1, 3)
= 5 * 3 + 2 * -1 + 4 * 3
= 15 - 2 + 12
= 25
Next, let's calculate |P0 - P| and |→d|:
|P0 - P| = sqrt(5^2 + 2^2 + 4^2)
= sqrt(25 + 4 + 16)
= sqrt(45)
|→d| = sqrt(19)
Using the formula mentioned earlier:
25 = sqrt(45) * sqrt(19) * cosθ
cosθ = 25 / (sqrt(45) * sqrt(19))
Now, we can substitute this value of cosθ into the equation of the line to find the point Q:
Q = P + t * →d
= (-4, -3, -5) + t * (3, -1, 3)
Since Q is on the line L, we can substitute the values of P and →d as:
Q = (-4, -3, -5) + t * (3, -1, 3)
= (-4 + 3t, -3 - t, -5 + 3t)
Therefore, the point Q on line L that is closest to P0 is Q = (-4 + 3t, -3 - t, -5 + 3t).
To find the shortest distance between a point and a line in 3D space, we can use the formula:
d = |(P0 - P) × →d| / |→d|
where P0 is the given point, P is a point on the line, →d is the direction vector of the line, "×" represents the cross product, and "|" denotes the magnitude (or length) of a vector.
1. Calculate the vector P0 - P:
P0 - P = (1, -1, -1) - (-4, -3, -5) = (1+4, -1+3, -1+5) = (5, 2, 4)
2. Calculate the cross product of (P0 - P) and →d:
(P0 - P) × →d = (5, 2, 4) × (3, -1, 3)
To calculate the cross product, we can use the formula:
(i) × (j) = (ik - jk, jk - ij, ij - ik)
3. Substituting the values into the formula:
(5, 2, 4) × (3, -1, 3) = (2 * 3 - 4 * -1, 4 * 3 - 5 * 3, 5 * -1 - 2 * 3)
= (6 + 4, 12 - 15, -5 - 6)
= (10, -3, -11)
4. Calculate the magnitude of (P0 - P) × →d:
|(P0 - P) × →d| = √(10^2 + (-3)^2 + (-11)^2)
= √(100 + 9 + 121)
= √(230)
5. Calculate the magnitude of →d:
|→d| = √(3^2 + (-1)^2 + 3^2)
= √(9 + 1 + 9)
= √(19)
6. Calculate the shortest distance d:
d = |(P0 - P) × →d| / |→d|
= √(230) / √(19)
= √(230/19)
Therefore,
d = √(230/19)
7. Calculate the point Q on the line that is closest to P0:
Q = P + ((P0 - P) · →d) / (→d · →d) * →d
To calculate the dot product, (P0 - P) · →d, we can use the formula:
(a, b, c) • (d, e, f) = ad + be + cf
Substituting the values into the formula:
(P0 - P) · →d = (5, 2, 4) • (3, -1, 3)
= 5 * 3 + 2 * -1 + 4 * 3
= 15 - 2 + 12
= 25
To calculate the dot product, →d · →d, we can use the formula:
(a, b, c) • (d, e, f) = ad + be + cf
Substituting the values into the formula:
→d · →d = (3, -1, 3) • (3, -1, 3)
= 3 * 3 + -1 * -1 + 3 * 3
= 9 + 1 + 9
= 19
Substituting the values into the formula to find Q:
Q = P + ((P0 - P) · →d) / (→d · →d) * →d
= (-4, -3, -5) + (25 / 19) * (3, -1, 3)
Calculating:
Q = (-4, -3, -5) + (25 / 19) * (3, -1, 3)
= (-4 + (25 / 19) * 3, -3 + (25 / 19) * -1, -5 + (25 / 19) * 3)
Therefore,
Q = (-4 + (25 / 19) * 3, -3 + (25 / 19) * -1, -5 + (25 / 19) * 3)