Let \(z\) be the complex number \(z=e^{\small{\dfrac{i \pi}{6}}} \) and \(w\) be the complex number \(w=e^{\small{\dfrac{3 i \pi}{4}}} \). --- 6 WORK AREA LINES (style=lined) --- --- 8 WORK AREA LINES (style=lined) --- --- 5 WORK AREA LINES (style=lined) ---
Vectors, EXT2 V1 2023 HSC 11b
Find the angle between the vectors
\(\underset{\sim}{a}=\underset{\sim}{i}+2 \underset{\sim}{j}-3 \underset{\sim}{k}\)
\(\underset{\sim}{b}=-\underset{\sim}{i}+4 \underset{\sim}{j}+2 \underset{\sim}{k}\),
giving your answer to the nearest degree. (3 marks)
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Vectors, EXT2 V1 EQ-Bank 12
Two planes are flying in directions given by the vectors `-200underset~i+400underset~j-3underset~k` and `300underset~i+250underset~j+underset~k`.
A person in the flight control centre is plotting their paths on a map.
Calculate the acute angle between their projected flight paths, giving your answer correct to one decimal place. (2 marks)
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Vectors, EXT2 V1 EQ-Bank 2
Find the angle between the vectors `underset~r = ((3),(-2),(-1))` and `underset~s = ((2),(1),(1))`, giving the angle in degrees correct to 1 decimal place. (3 marks)
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Vectors, EXT2 V1 2022 HSC 11d
A triangle is formed in three-dimensional space with vertices `A(1,-1,2)`, `B(0,2,-1)` and `C(2,1,1)`.
Find the size of `/_ABC`, giving your answer to the nearest degree. (3 marks)
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Vectors, EXT2 V1 2021 HSC 11c
Find the angle between the vectors `underset~a = ((2),(0),(4))` and `underset~b = ((-3),(1),(2))`, giving the angle in degrees correct to 1 decimal place. (3 marks)
Vectors, EXT2 V1 2020 SPEC2 16 MC
Let `underset~a = underset~i + 2underset~j + 2underset~k` and `underset~b = 2underset~i - 4underset~j + 4underset~k`, where the acute angle between these vectors is `theta`.
The value of `sin(2theta)` is
- `1/9`
- `(4sqrt5)/9`
- `(4sqrt5)/81`
- `(8sqrt5)/81`
Vectors, EXT2 V1 SM-Bank 25
The acute angle `theta` is the angle between the vectors `underset~a = −2underset~i + 2underset~j - underset~k` and `underset~b = −4underset~i + 4underset~j + 7underset~k`.
Find the exact value of `sin(2theta)`. (2 marks)
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Vectors, EXT2 V1 2017 SPEC1 5
Relative to a fixed origin, the points `B`, `C` and `D` are defined respectively by the position vectors `underset~b = underset~i - underset~j + 2underset~k, \ underset~c = 2underset~i - underset~j + underset~k` and `underset~d = aunderset~i - 2underset~j` where `a` is a real constant.
Given that the magnitude of angle `BCD` is `pi/3`, find `a`. (4 marks)
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Vectors, EXT2 V1 2017 SPEC1 10
Consider the vectors `underset ~a = - underset ~i - 2 underset ~j + 3 underset ~k` and `underset ~b = 2 underset ~i + c underset ~j + underset ~k`.
Find the value of `c, \ c in R`, if the angle between `underset ~a` and `underset ~b` is `pi/3`. (4 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 SM-Bank 9
Points `A`, `B` and `C` have position vectors `underset~a = 2underset~i + underset~j`, `underset~b = 3underset~i - underset~j + underset~k` and `underset~c = -3underset~j + underset~k` respectively.
Find the cosine of angle `ABC`. (2 marks)
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Vectors, EXT2 V1 SM-Bank 7
If `theta` is the angle between `underset ~a = sqrt 3 underset ~i + 4 underset ~j - underset ~k` and `underset ~b = underset ~i - 4 underset ~j + sqrt 3 underset ~k`, then find `cos(2 theta)`. (2 marks)
Vectors, EXT2 V1 2011 SPEC2 12 MC
The angle between the vectors `3underset~i + 6underset~j - 2underset~k` and `2underset~i - 2underset~j + underset~k`, correct to the nearest tenth of a degree, is
A. 2.0°
B. 91.0°
C. 112.4°
D. 121.3°