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Vectors, EXT2 V1 2023 HSC 5 MC

Which of the following is a true statement about the lines  \(\ell_1={\displaystyle\left(\begin{array}{cc}-1 \\ 2 \\ 5\end{array}\right)+\lambda\left(\begin{array}{c}-1 \\ 3 \\ 1\end{array}\right)}\)  and  \(\ell_2=\left(\begin{array}{c}3 \\ -10 \\ 1\end{array}\right)+\mu\left(\begin{array}{c}1 \\ -3 \\ -1\end{array}\right) ?\)

  1. \(\ell_1\) and \(\ell_2\) are the same line.
  2. \(\ell_1\) and \(\ell_2\) are not parallel and they intersect.
  3. \(\ell_1\) and \(\ell_2\) are parallel and they do not intersect.
  4. \(\ell_1\) and \(\ell_2\) are not parallel and they do not intersect.
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\(A\)

Show Worked Solution

\(\text{Since}\ \ \left(\begin{array}{c}-1 \\ 3 \\ 1\end{array}\right) = -1 \left(\begin{array}{c}1 \\ -3 \\ -1\end{array}\right), \ \ell_1\ \text{is parallel to}\ \ell_2 \)

\(\text{Test if point}\ (3,-10,1)\ \text{lies on}\ \ell_1: \)

\(\text{i.e.}\ \ \exists \lambda\ \ \text{such that} \)

♦ Mean mark 49%.

\( \left(\begin{array}{cc}3 \\ -10 \\ 1\end{array}\right) = \left(\begin{array}{cc}-1 \\ 2 \\ 5\end{array}\right) + \lambda \left(\begin{array}{cc}-1 \\ 3 \\ 1\end{array}\right)\)

\(\lambda = -4\ \ \text{satisfies equation} \)

\(\therefore\ \ell_1\ \text{and}\ \ell_2\ \text{are the same line.}\)

\(\Rightarrow A\)

Vectors, EXT2 V1 2022 HSC 11e

Let `ℓ_(1)` be the line with equation `([x],[y])=([-1],[7])+lambda([3],[2]),lambda inRR`.

The line `ℓ_(2)` passes through the point  `A(-6,5)`  and is parallel to `ℓ_(1)`.

Find the equation of the line `ℓ_(2)` in the form  `y=mx+c`.  (2 marks)

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`y=2/3x+9`

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`m_(ℓ_(1))=2/3`

`text{Equation of}\ ℓ_(2)\ text{has}\ m=2/3\ text{and passes through}\ (-6,5):`

`y-5` `=2/3(x+6)`  
`y` `=2/3x+9`  

Vectors, EXT2 V1 EQ-Bank 18

Determine the equation of the line vector `underset~r`, given it passes through the point `(7, 1, 0)` and is parallel to the line joining `P(2, −1, 2)` and `Q(3, 4, 1)`.   (2 marks)

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`underset~r = ((7),(1),(0)) + lambda((1),(5),(−1))`

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`overset(->)(PQ) = ((3),(4),(1)) – ((2),(−1),(2)) = ((1),(5),(−1))`
 

`:. underset~r = ((7),(1),(0)) + lambda((1),(5),(−1))`

Vectors, EXT2 V1 EQ-Bank 13

Consider the vectors  `underset~u = a underset~i - b underset~j + c underset~k`  and  `underset~v = underset~i - 8underset~j + 4underset~k`.

Find all possible values of `a, b` and `c` if `underset~u` is parallel to `underset~v` and  has a magnitude of 3.   (3 marks)

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`1/3 , 8/3 , 4/3`

`text(or)`

` -1/3 , – 8/3 , – 4/3`

Show Worked Solution
`|underset~v|` `= sqrt(1 + 64 + 16) = 9`
`underset~overset^v` `= underset~v /|underset~v| =  (1)/(9) underset~i – (8)/(9) underset~j + (4)/(9) underset~k \ \ text{(magnitude of 1)}`

 
`text(S) text(ince) \ underset~u  \ text(has a magnitude of 3:)`

`underset~u` `= ± 3 ((1)/(9) underset~i – (8)/(9) underset~j + (4)/(9) underset~k)`
  `= ± (1/3 underset~i – 8/3 underset~j + 4/3 underset~k)`

 

`:. \ a, b, c` `= 1/3 , – 8/3 , 4/3\ \ text{or}\ \ -1/3 , 8/3 , – 4/3`