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Vectors, EXT1 V1 EQ-Bank 6 MC

The diagram below, \(ABCD\), is a quadrilateral. The origin, \(O\), is not shown.
 

Which of the following is true?

  1. \(\overrightarrow{AC}=\overrightarrow{OA}-\overrightarrow{OC}\)
  2. \(\overrightarrow{DB}=\overrightarrow{DA}-\overrightarrow{AB}\)
  3. \(\overrightarrow{DB}=\overrightarrow{OB}-\overrightarrow{OD}\)
  4. \(\overrightarrow{AC}=\overrightarrow{OC}+\overrightarrow{OA}\)
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\(\Rightarrow C\)

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\(\text{The origin can be located anywhere.}\)

\(\text{By definition:}\)

\(\overrightarrow{DB}=\overrightarrow{OB}-\overrightarrow{OD}\)

\(\Rightarrow C\)

Vectors, EXT1 V1 SM-Bank 31

The sum of two unit vectors is a unit vector.

Determine the magnitude of the difference of the two vectors.   (3 marks)

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\(D\)

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\(\text{Vectors can be drawn as two sides of an equilateral triangle.}\)

\(\text{Using the cosine rule for the difference between the two vectors:}\)

\(c^2\) \(=a^2+b^2-2ab\, \cos C\)  
  \(=1+1-2\times -\dfrac{1}{2} \)  
  \(=3\)  
\(c\) \(=\sqrt{3}\)  

 
\(\abs{v_1-v_2} = \sqrt{3}\)

Vectors, EXT1 V1 2021 SPEC2 11

Let `underset~i` be a unit vector pointing east and let `underset~j` be a unit vector pointing north.

A group of hikers travels 5 km in the direction south 30° west and then north for 10 km before stopping for lunch at position vector  `(a underset~i, b underset~j)`.

Find the values of `a` and `b`.   (2 marks)

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`a=-5/2, b=10-(5sqrt3)/2`

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`sin30 = x/5 \ => \ x = 5/2`

`cos30 = y/5 \ => \ y = (5sqrt3)/2`

`P_1\ text(vector:)\  (-5/2 underset~i, -(5sqrt3)/2 underset~j)`

`P_2\ text(vector:)\  (-5/2 underset~i, (10-(5sqrt3)/2)underset~j)`

`:. a=-5/2, b=10-(5sqrt3)/2`

Vectors, EXT1 V1 2020 HSC 9 MC

The projection of the vector  `((6),(7))`  onto the line  `y = 2x`  is  `((4),(8))`.

The point  `(6, 7)`  is reflected in the line  `y = 2x`  to a point `A`.

What is the position vector of the point `A`?

  1. `((6),(12))`
  2. `((2),(9))`
  3. `((−6),(7))`
  4. `((−2),(1))`
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`B`

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`text(Graph the projection and reflection:)`

 

`=>B`

Vectors, EXT1 V1 2020 HSC 4 MC

Maria starts at the origin and walks along all of the vector  `2underset~i + 3underset~j`, then walks along all of the vector  `3underset~i - 2underset~j`  and finally along all of the vector  `4underset~i - 3underset~j`.

How far from the origin is she?

  1. `sqrt77`
  2. `sqrt85`
  3. `2sqrt13 + sqrt5`
  4. `sqrt5 + sqrt7 + sqrt13`
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`B`

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`underset~v` `= ((2),(3)) + ((3),(−2)) + ((4),(−3))`
  `= ((9),(−2))`
`|underset~v|` `= sqrt(9^2 + (−2)^2)`
  `= sqrt85`

 
`=>B`

Vectors, EXT1 V1 EQ-Bank 4 MC

The diagram shows a grid of equally spaced lines. The vector  `overset(->)(OA) = underset~a`  and the vector  `overset(->)(OH) = underset~h`. The point `Q` is halfway between `F` and `H`.
 

Which expression represents the vector `overset(->)(EQ)`?

  1. `−1/4 underset~a + 1/2 underset~h`
  2. `1/2 underset~a - 1/4 underset~h`
  3. `1/4 underset~a + 1/2 underset~h`
  4. `1/4 underset~a + underset~h`
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`A`

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`overset(->)(EQ) = 1/2underset~h-1/4underset~a`

`=> A`