Prove that the vectors `4 underset ~i + 5 underset ~j - 2 underset ~k` and ` −5 underset ~i + 6 underset ~j + 5underset ~k`, are perpendicular. (2 marks)
Vectors, EXT2 V1 2020 HSC 11d
Consider the two vectors `underset~u = 2 underset~i - underset~j + 3 underset~k` and `underset~v = p underset~i + underset~j + 2 underset~k`.
For what values of `p` are `underset~u - underset~v` and `underset~u + underset~v` perpendicular? (3 marks)
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Vectors, EXT2 V1 2014 SPEC1 1
Consider the vector `underset ~a = sqrt 3 underset ~i - underset ~j - sqrt 2 underset ~k`, where `underset ~i, underset ~j` and `underset ~k` are unit vectors in the positive directions of the `x, y` and `z` axes respectively.
- Find the unit vector in the direction of `underset ~a`. (1 mark)
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- Find the acute angle that `underset ~a` makes with the positive direction of the `x`-axis. (2 marks)
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- The vector `underset ~b = 2 sqrt 3 underset ~i + m underset ~j - 5 underset ~k`.
Given that `underset ~b` is perpendicular to `underset ~a,` find the value of `m`. (2 marks)
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Vectors, EXT2 V1 2013 SPEC1 3
The coordinates of three points are `A\ ((– 1), (2), (4)), \ B\ ((1), (0), (5)) and C\ ((3), (5), (2)).`
- Find `vec (AB).` (1 mark)
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- The points `A, B` and `C` are the vertices of a triangle.
Prove that the triangle has a right angle at `A.` (2 marks)
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- Find the length of the hypotenuse of the triangle. (1 mark)
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Vectors, EXT2 V1 SM-Bank 8
If `underset ~a = -2 underset ~i - underset ~j + 3 underset ~k` and `underset ~b = -m underset ~i + underset ~j + 2 underset ~k`, where `m` is a real constant, find `m` such that the vector `underset ~a - underset ~b` will be perpendicular to vector `underset ~b`. (2 marks)
Vectors, EXT2 V1 SM-Bank 12
Two vectors are given by `underset ~a = 4 underset ~i + m underset ~j - 3 underset ~k` and `underset ~b = −2 underset ~i + n underset ~j - underset ~k`, where `m`, `n in R^+`.
If `|\ underset ~a\ | = 10` and `underset ~a` is perpendicular to `underset ~b`, then find `m` and `n`. (2 marks)
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Vectors, EXT2 V1 SM-Bank 5
Find the value(s) of `m` so that the vectors `underset~a = 2underset~i + m underset~j - 3underset~k` and `underset~b = m^2underset~i - underset~j + underset~k` are perpendicular. (2 marks)
Vectors, EXT2 V1 SM-Bank 4
Consider the three vectors
`underset ~a = underset ~i - underset ~j + 2underset ~k,\ underset ~b = underset ~i + 2 underset ~j + m underset ~k` and `underset ~c = underset ~i + underset ~j - underset ~k`, where `m in R.`
- Find the value(s) of `m` for which `|\ underset ~b\ | = 2 sqrt 3.` (2 marks)
- Find the value of `m` such that `underset ~a` is perpendicular to `underset ~b.` (1 mark)