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Vectors, SPEC2 2013 VCAA 15 MC

Let  `underset~u = 4underset~i - underset~j + underset~k`,  `underset~v = 3underset~j + 3underset~k`  and  `underset~w = −4underset~i + underset~j + underset~k`.

Which one of the following statements is not true?

  1. `|\ underset~u\ | = |\ underset~v\ |`
  2. `|\ underset~u\ | = |\ −underset~w\ |`
  3. `underset~u`, `underset~v` and `underset~w` are linearly independent
  4. `underset~u.underset~v = 0`
  5. `(underset~u + underset~w).underset~v = 12`
Show Answers Only

`E`

Show Worked Solution
`|underset~u|` `= sqrt(16 + 1 + 1)=sqrt18`
`|underset~v|` `= sqrt(9 + 9)=sqrt18`
`|underset~w|` `= sqrt(16+1+1)=sqrt18`

 
`underset~u · underset~v= 4 xx 0 + (−1) xx 3 + 1 xx 3=0`
 

`(underset~u + underset~w) · underset~v` `= ((4 – 4)underset~i + (−1 + 1)underset~j + (1 + 1)underset~k) · underset~v`
  `= 2underset~k · underset~v`
  `= 6`

 
`=> E`

Vectors, SPEC2 2018 VCAA 12 MC

If  `|underset ~a + underset ~b| = |underset ~a| + |underset ~b|`  and  `underset ~a, underset ~b != underset ~0`, which one of the following is necessarily true?

  1. `underset ~a\ text(is parallel to)\ underset ~b`
  2. `|underset ~a| = |underset ~b|`
  3. `underset ~a = underset ~b`
  4. `underset ~a = -underset ~b`
  5. `underset ~a\ text(is perpendicular to)\ underset ~b` 
Show Answers Only

`A`

Show Worked Solution
`|underset ~a + underset ~b|^2` `= (|underset ~a| + |underset ~b|)^2\ \ \ text{(given)}`
  `= |underset ~a|^2 + 2|underset ~a||underset ~b|+|underset ~b|^2`
`underset ~a ⋅ underset ~b` `= |underset ~a||underset ~b| cos theta`

 
`=> 2|underset ~a||underset ~b| = (2 underset ~a ⋅ underset ~b)/(cos theta)`

♦♦ Mean mark 36%.

`=>|underset ~a + underset ~b|^2 = |underset ~a|^2 + (2 underset ~a ⋅ underset ~b)/(cos theta) + |b|^2`

`(underset ~a + underset ~b) * (underset ~a + underset ~b) = underset ~a ⋅ underset ~a + (2 underset ~a ⋅ underset ~b)/(cos theta) + underset ~b ⋅ underset ~b`

`underset ~a ⋅ underset ~a + 2underset ~a ⋅ underset ~b + underset ~b ⋅ underset ~b = underset ~a ⋅ underset ~a + (2 underset ~a ⋅ underset ~b)/(cos theta) + underset ~b ⋅ underset ~b`

`2 underset ~a ⋅ underset ~b = (2 underset ~a ⋅ underset ~b)/(cos theta)`

`:. cos theta = 1\ \ =>\ \  theta = 0`

`=>  A`