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5. (a) Show that the points A(1,5,–3), B(4,–1,0) and C(8,–9,4) are collinear - Scottish Highers Maths - Question 5 - 2022
Question 5
5. (a) Show that the points A(1,5,–3), B(4,–1,0) and C(8,–9,4) are collinear. (b) State the ratio in which B divides AC.
Worked Solution & Example Answer:5. (a) Show that the points A(1,5,–3), B(4,–1,0) and C(8,–9,4) are collinear - Scottish Highers Maths - Question 5 - 2022
Step 1
Show that the points A(1,5,–3), B(4,–1,0) and C(8,–9,4) are collinear.
Answer
To show that the points are collinear, we can use the concept of vectors. First, we will find the vectors AB and BC:
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Calculate vector AB:
AB = B - A = (4 - 1, -1 - 5, 0 - (-3)) = (3, -6, 3)
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Calculate vector BC:
BC = C - B = (8 - 4, -9 - (-1), 4 - 0) = (4, -8, 4)
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Determine if AB and BC are parallel:
For AB and BC to be parallel, there must exist a scalar k such that:
AB = k * BC
This means:
(3, -6, 3) = k * (4, -8, 4)
Solving for k:
- From the first component: 3 = 4k → k = 3/4
- From the second component: -6 = -8k → k = 3/4
- From the third component: 3 = 4k → k = 3/4
Since k is consistent across all components, we confirm:
AB is parallel to BC. Thus, points A, B, and C are collinear.
Step 2
State the ratio in which B divides AC.
Answer
We can find the ratio in which B divides AC by using the section formula. The coordinates of A and C are:
- A(1, 5, -3)
- C(8, -9, 4)
Using the formula for the ratio of division for coordinates:
If B divides AC in the ratio m:n, then:
B = rac{nA + mC}{m + n}
Substituting the values:
B = B(4, -1, 0) = rac{n(1, 5, -3) + m(8, -9, 4)}{m + n}
Solving for m:n, we find that B divides AC in the ratio 3:4.