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Vectors a and b are such that a = 4i - 2j + 2k and b = -2i + j + pk - Scottish Highers Maths - Question 12 - 2018
Question 12
Vectors a and b are such that a = 4i - 2j + 2k and b = -2i + j + pk. (a) Express 2a + b in component form. (b) Hence find the values of p for which |2a + b| = 7.
Worked Solution & Example Answer:Vectors a and b are such that a = 4i - 2j + 2k and b = -2i + j + pk - Scottish Highers Maths - Question 12 - 2018
Step 1
Express 2a + b in component form.
Answer
To express the vector (2a + b) in component form, we first calculate (2a) and then add vector (b).
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Calculate (2a:)
[2a = 2(4i - 2j + 2k) = 8i - 4j + 4k.]
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The vector (b) is given as (b = -2i + j + pk.]
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Now, add (2a) and (b):
[2a + b = (8i - 4j + 4k) + (-2i + j + pk) = (8 - 2)i + (-4 + 1)j + (4 + p)k.]
Thus, in component form: [2a + b = 6i - 3j + (4 + p)k.]
Step 2
Hence find the values of p for which |2a + b| = 7.
Answer
To find the values of (p) such that (|2a + b| = 7), we first calculate the magnitude of the vector (2a + b):
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The magnitude is given by:
[|2a + b| = \sqrt{(6)^2 + (-3)^2 + (4 + p)^2}.]
This simplifies to:
[|2a + b| = \sqrt{36 + 9 + (4 + p)^2} = \sqrt{45 + (4 + p)^2}.]
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Set the magnitude equal to 7:
[\sqrt{45 + (4 + p)^2} = 7.]
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Square both sides:
[45 + (4 + p)^2 = 49.]
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Subtract 45 from both sides:
[(4 + p)^2 = 4.]
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Take square root of both sides:
[4 + p = 2 \quad \text{or} \quad 4 + p = -2.]
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Solve for p:
- From (4 + p = 2): (p = 2 - 4 = -2.)
- From (4 + p = -2): (p = -2 - 4 = -6.)
Thus, the values of (p) are (p = -2) and (p = -6).