a) The solution of the system is: (λ, a, b) = (- 5, - 11, - 26).
b) The coordinates of the point P are (x, y, z) = (- 16 / 3, - 43, 152 / 3).
c) Vectors <u>AP</u> and <u>PB</u> are <em>collinear</em> for their ends lie on the line. The ratio AP : PB is 83.248 : 73.491 (1.133 : 1).
<h3>How to analyze a line equation in vector form</h3>
a) In this problem we have a equation of the line in <em>vector</em> form, whose form is:
<u>r</u> = (5, 19, - 1) + λ · (1, 6, - 5)
Now we find the value of λ for the case <u>r</u> = (0, a, b):
(0, a, b) = (5, 19, - 1) + λ · (1, 6, - 5)
5 + λ = 0
19 + 6 · λ = a
- 1 + 5 · λ = b
The solution of the system is: (λ, a, b) = (- 5, - 11, - 26).
b) Then, the following condition must be fulfilled:
<u>OP</u> • <u>PB</u> = 0
(5 + λ, 19 + 6 · λ, - 1 + 5 · λ) • [(4, 13, 4) - (5 + λ, 19 + 6 · λ, - 1 - 5 · λ)] = 0
(5 + λ, 19 + 6 · λ, - 1 + 5 · λ) • (- 1 - λ, - 6 - 6 · λ, 5 + 5 · λ) = 0
(5 + λ) · (- 1 - λ) + (19 + 6 · λ) · (- 6 - 6 · λ) + (- 1 + 5 · λ) · (5 + 5 · λ) = 0
- (5 + λ) · (1 + λ) - 6 · (19 + 6 · λ) · (1 + λ) + 5 · (- 1 + 5 · λ) · (1 + λ) = 0
- (5 + 6 · λ + λ²) - 6 · (19 + 25 · λ + 6 · λ²) + 5 · (- 1 + 4 · λ + 5 · λ²) = 0
- 5 - 6 · λ - λ² - 114 - 150 · λ - 36 · λ² - 5 + 20 · λ + 25 · λ² = 0
- 12 · λ² - 136 · λ - 124 = 0
There are two solutions: λ₁ = - 1, λ₂ = - 31 / 3, whose points P are:
<u>r₁</u> = (5, 19, - 1) + (- 1) · (1, 6, - 5)
<u>r₁</u> = (4, 13, 4)
<u>r₂</u> = (5, 19, - 1) + (- 31 / 3) · (1, 6, - 5)
<u>r₂</u> = (- 16 / 3, - 43, 152 / 3)
Then, the coordinates of the point P are (x, y, z) = (- 16 / 3, - 43, 152 / 3).
c) Vectors <u>AP</u> and <u>PB</u> are <em>collinear</em> for their ends lie on the line. The ratio can be found by using the magnitudes of each vector:
Vector AP
<u>AP</u> = (- 16 / 3, - 43, 152 / 3) - (0, - 11, - 26)
<u>AP</u> = (- 16 / 3, - 32, 230 / 3)
AP = √(<u>AP</u> • <u>AP</u>)
AP ≈ 83.248
Vector PB
<u>PB</u> = (4, 13, 4) - (- 16 / 3, - 43, 152 / 3)
<u>PB</u> = (28 / 3, 56, - 140 / 3)
PB = √(<u>PB</u> • <u>PB</u>)
PB ≈ 73.491
The ratio AP : PB is 83.248 : 73.491 (1.133 : 1).
To learn more on lines: brainly.com/question/17188072
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