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Vectors, EXT1 2017 NHT 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` if the angle between `underset ~a` and `underset ~b` is `pi/3`.   (4 marks)

Show Answers Only

`c = -3`

Show Worked Solution
`underset ~a ⋅ underset ~b` `= -1 xx 2 + (-2) xx c + 3 xx 1`
  `= -2-2c + 3`
  `= 1-2c`

 

`1-2c` `= sqrt((-1)^2 + (-2)^2 + 3^3) *sqrt(2^2 + c^2 + 1^2) xx cos (pi/3)`
`1-2c` `= 1/2(sqrt 14 ⋅ sqrt(5 + c^2))`
`2-4c` `= sqrt(14(5 + c^2))`
`(2-4c)^2` `= 14(5 + c^2)`
`4-16c + 16c^2` `= 70 + 14c^2`
`2c^2-16c-66` `= 0`
`c^2-8c-33` `= 0`
`(c-11)(c + 3)` `= 0`

 
`c = 11 or c = -3`

`text(S)text(ince)\ \ 2-4c = sqrt(15(5 + c^2))`

`2-4c > 0\ \ =>\ \ c<2`

`:. c = -3`

Vectors, EXT1 2024 SPEC1 4

Consider the vectors  \(\underset{\sim}{ a }=3 \underset{\sim}{ j }+3 \underset{\sim}{ k }\)  and  \(\underset{\sim}{ b }=2 \underset{\sim}{ i }-\underset{\sim}{ j }-2 \underset{\sim}{ k }\).

Find the angle between \(\underset{\sim}{ a }\) and \(\underset{\sim}{ b }\).   (2 marks)

Show Answers Only

\(\theta=\dfrac{3 \pi}{4}\left(\text{or} \ 135^{\circ}\right)\)

Show Worked Solution

\(\underset{\sim}{a}=\left(\begin{array}{l}0 \\ 3 \\ 3\end{array}\right) \Rightarrow \abs{\underset{\sim}{a}}=\sqrt{18}=3 \sqrt{2}\)

     \(\underset{\sim}{b}=\left(\begin{array}{c}2 \\ -1 \\ -2\end{array}\right) \Rightarrow\abs{\underset{\sim}{b}}=\sqrt{9}=3\)

     \(\cos \theta=\dfrac{\underset{\sim}{a} \cdot \underset{\sim}{b}}{\abs{\underset{\sim}{a}} \cdot \abs{\underset{\sim}{b}}}=\dfrac{-3-6}{3 \sqrt{2} \times 3}=-\dfrac{1}{\sqrt{2}}\)

    \(\therefore \theta=\cos ^{-1}\left(-\dfrac{1}{\sqrt{2}}\right)=\dfrac{3 \pi}{4}\left(\text{or} \ 135^{\circ}\right)\)

Vectors, EXT1 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.

  1. Find the unit vector in the direction of  `underset ~a`.   (1 mark)

  2. Find the acute angle that `underset ~a` makes with the positive direction of the `x`-axis.   (2 marks)

  3. The vector  `underset ~b = 2 sqrt 3 underset ~i + m underset ~j-5 underset ~k`.
  4. Given that `underset ~b` is perpendicular to `underset ~a,` find the value of `m`.  (2 marks)

Show Answers Only

a.    `1/sqrt 6 (sqrt 3 underset ~i-underset ~j-sqrt 2 underset ~k)`

b.    `theta = 45^@`

c.    `m = 6 + 5 sqrt 2`

Show Worked Solution

a.    `|underset ~a|= sqrt((sqrt 3)^2 + (-1)^2 + (-sqrt 2)^2)= sqrt 6`

`hat underset ~a= underset ~a/|underset ~a|= 1/sqrt 6 (sqrt 3 underset ~i-underset ~j-sqrt 2 underset ~k)`
 

b.    `x text{-axis vectors include}\ (1,0,0).`

`underset ~a ⋅ underset ~i = ((\sqrt3),(-1),(-\sqrt2))((1),(0),(0))=\sqrt3`

  `underset ~a ⋅ underset ~i` `= |underset ~a||underset ~i| cos theta= sqrt 6 cos theta`
  `sqrt 3` `= sqrt 6 cos theta`
  `cos theta` `=1/sqrt 2`
  `:. theta` `= 45^@`

 
c.   `underset ~a ⋅ underset ~b = sqrt 3 (2 sqrt 3) + (-1)(m) + (-sqrt 2)(-5) = 0`

`6-m + 5 sqrt 2` `=0`  
`:. m` `=6 + 5 sqrt 2`  

Vectors, EXT1* V1 2024 HSC 11c

Find the angle between the two vectors  \(\underset{\sim}{u}=\left(\begin{array}{c}1 \\ 2 \\ -2\end{array}\right)\) and  \(\underset{\sim}{v}=\left(\begin{array}{c}4 \\ -4 \\ 7\end{array}\right)\), giving your answer in radians, correct to 1 decimal place.   (2 marks)

Show Answers Only

\(\theta=2.3^c \ \ \text{(1 d.p.)}\)

Show Worked Solution

\(\underset{\sim}{u}=\left(\begin{array}{c}1 \\ 2 \\ -2\end{array}\right),\abs{\underset{\sim}{u}}=\sqrt{1+4+4}=3\)

\(\underset{\sim}{v}=\left(\begin{array}{c}4 \\ -4 \\ 7\end{array}\right),\abs{\underset{\sim}{v}}=\sqrt{16+16+49}=9\)

\(\cos \theta=\dfrac{\underset{\sim}{u} \cdot \underset{\sim}{v}}{|\underset{\sim}{u}||\underset{\sim}{v}|}=\dfrac{1 \times 4-2 \times 4-2 \times 7}{3 \times 9}=-\dfrac{2}{3}\)

\(\theta=\cos ^{-1}\left(-\dfrac{2}{3}\right)=2.30 \ldots=2.3^c \ \ \text{(1 d.p.)}\)

Vectors, EXT1* 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)

Show Answers Only

`33°`

Show Worked Solution

`vec(BA)=((1),(-1),(2))-((0),(2),(-1))=((1),(-3),(3))`

`abs(vec(BA))=sqrt(1^2+3^2+3^2)=sqrt19`
 

`vec(BC)=((2),(1),(1))-((0),(2),(-1))=((2),(-1),(2))`

`abs(vec(BC))=sqrt(2^2+1^2+2^2)=sqrt9=3`
 

`vec(BA)*vec(BC)=1xx2+ -3xx-1+3xx2=11`

`cos/_ABC=(vec(BA)*vec(BC))/(abs{vec(BA)}abs{vec(BC)})=11/(3sqrt19)`

`:./_ABC=cos^(-1)(11/(3sqrt19))=32.733…=33°\ \ text{(nearest degree)}`

Vectors, EXT1* 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)

Show Answers Only

\(87^{\circ} \)

Show Worked Solution

\[\underset{\sim}{a}=\left(\begin{array}{c} 1 \\ 2 \\ -3 \end{array}\right),\ \  \underset{\sim}{b}=\left(\begin{array}{c} -1 \\ 4 \\ 2 \end{array}\right) \]

\(\Big{|} \underset{\sim}{a} \Big{|} = \sqrt{1+4+9} = \sqrt{14} \)

\(\Big{|} \underset{\sim}{b} \Big{|} = \sqrt{1+16+4} = \sqrt{21} \)

\( \underset{\sim}{a} \cdot \underset{\sim}{b} = -1 + 8-6=1 \)

\(\cos\ \theta \) \(=\dfrac{\underset{\sim}{a} \cdot \underset{\sim}{b}}{\Big{|}\underset{\sim}{a}\Big{|} \cdot \Big{|}\underset{\sim}{b}\Big{|}} \)  
  \(=\dfrac{1}{\sqrt{294}} \)  
\( \theta\) \(=\cos ^{-1} \Big{(}\dfrac{1}{\sqrt{294}}\Big{)} \)  
  \(=86.65…\)  
  \(=87^{\circ} \)  

Vectors, EXT1* 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)

Show Answers Only

`83.1^@`

Show Worked Solution

`underset~a = ((2),(0),(4)) \ , \ |underset~a| \ = sqrt{2^2 + 4^2} = sqrt20`

`underset~b = ((-3),(1),(2)) \ , \ |underset~b| \ = sqrt{(-3)^2 + 1^2 + 2^2} = sqrt14`

`underset~a * underset~b` `= ((2),(0),(4)) ((-3),(1),(2)) = – 6 + 0 + 8 = 2`
`underset~a * underset~b` `= |underset~a| |underset~b| \ cos theta`
`2` `= sqrt20 sqrt14 \ cos theta`
`cos theta` `= 2/sqrt280`
`theta` `= cos^(-1) (1/sqrt70)`
  `= 83.1^@ \ text{(1 d.p,)}`

Vectors, EXT1 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

  1. 2.0°
  2. 91.0°
  3. 112.4°
  4. 121.3°
Show Answers Only

`C`

Show Worked Solution

`|3underset~i + 6underset~j-2underset~k| = sqrt(9 + 36 + 4) = sqrt49 = 7`

`|2underset~i-2underset~j + underset~k| = sqrt(4 + 4 + 1) = sqrt9 = 3`

`(3underset~i + 6underset~j-2underset~k) * (2underset~i-2underset~j + underset~k)`

`= 3 xx 2 + 6 xx (−2) + (−2) xx 1`

`= 6-12-2`

`= -8`  

`costheta` `= ((3tildei + 6tildej-2tildek).(2tildei-2tildej + tildek))/(|\ 3tildei + 6tildej-2tildek\ ||\ 2tildei-2tildej + tildek\ |)= -8/21`
`:. theta `= cos^(−1)(−8/12)~~ 112.4^@`

 
`=> C`

Vectors, EXT1 V1 2025 HSC 9 MC

The vectors \(\underset{\sim}{a}, \underset{\sim}{b}\) and \(\underset{\sim}{c}\) have magnitudes 3, 5 and 7 respectively.
 

Given that \(\underset{\sim}{a}+\underset{\sim}{b}+\underset{\sim}{c}=\underset{\sim}{0}\), what is the size of angle \(\theta\) between \(\underset{\sim}{a}\) and \(\underset{\sim}{b}\) ?

  1. \(\dfrac{\pi}{6}\)
  2. \(\dfrac{\pi}{3}\)
  3. \(\dfrac{2 \pi}{3}\)
  4. \(\dfrac{5 \pi}{6}\)
Show Answers Only

\(B\)

Show Worked Solution

\(\text{Find}\ \theta\ \text{using:}\ \ \cos\,\theta = \dfrac{\underset{\sim}{a} \cdot \underset{\sim}{b}}{\abs{\underset{\sim}{a}} \, \abs{\underset{\sim}{b}}}\)

\(\underset{\sim}{a}+\underset{\sim}{b}+\underset{\sim}{c}=0 \ \Rightarrow\  \underset{\sim}{a}+\underset{\sim}{b}=-\underset{\sim}{c}\)

\(\abs{\underset{\sim}{a}+\underset{\sim}{b}}=\abs{-\underset{\sim}{c}}=7\)

♦ Mean mark 45%.
\(\abs{\underset{\sim}{a}+\underset{\sim}{b}}^2\) \(=(\underset{\sim}{a}+\underset{\sim}{b})(\underset{\sim}{a}+\underset{\sim}{b})=49\)
\(49\) \(=\underset{\sim}{a} \cdot \underset{\sim}{a}+2 a \cdot \underset{\sim}{b}+\underset{\sim}{b} \cdot \underset{\sim}{b}\)
\(49\) \(=\abs{\underset{\sim}{a}}^2+2 a \cdot b+\abs{\underset{\sim}{b}}^2\)
\(49\) \(=9+2 \underset{\sim}{a} \cdot \underset{\sim}{b}+25\)
\(2 \underset{\sim}{a} \cdot \underset{\sim}{b}\) \(=15\)
\(\underset{\sim}{a} \cdot \underset{\sim}{b}\) \(=\dfrac{15}{2}\)

\(\cos \theta\) \(=\dfrac{\frac{15}{2}}{3 \times 5}=\dfrac{1}{2}\)
\(\therefore \theta\) \(=\dfrac{\pi}{3}\)

 
\(\Rightarrow B\)

Vectors, EXT1 V1 2025 HSC 11e

For what value of \(m\) is the vector \(\displaystyle \binom{1}{m}\) parallel to the vector \(\displaystyle \binom{2}{6}\)?   (1 mark)

Show Answers Only

\(m=3\)

Show Worked Solution

\(\text{If vectors are parallel:}\)

\(\displaystyle \binom{2}{6}=k\binom{1}{m} \ \Rightarrow \ k=2\)

\(2m\) \(=6\)
\(m\) \(=3\)

Vectors, EXT1 V1 2022 HSC 8 MC

The angle between two unit vectors `underset~a` and `underset~b` is `theta` and  `|underset~a+ underset~b| < 1`.

Which of the following best describes the possible range of values of  `theta` ?

  1. `0 <= theta < (pi)/(3)`
  2. `0 <= theta < (2pi)/(3)`
  3. `(pi)/(3) < theta <= pi`
  4. `(2pi)/(3) < theta <= pi`
Show Answers Only

`D`

Show Worked Solution

`text{By Elimination:}`

`text{Consider}\ \ underset~a=((1),(0)) and underset~b=((-1),(0))`

`|underset~a+ underset~b| =0 < 1\ \ and\ \ theta=pi`

`text{→ Eliminate A and B}`
 

`text{Consider}\ \ theta=(2pi)/3 and underset~a=((1),(0)),\ \ underset~b=((costheta),(sintheta))`

`underset~a+underset~b=((1+cos((2pi)/3)),(sin((2pi)/3)))=((1/2),(sqrt3/2))`

`|underset~a+ underset~b|^2 = (1/2)^2+(sqrt3/2)^2=1`

`:. theta !=(2pi)/3`

`text{→ Eliminate C}`

`=>D`


♦♦ Mean mark 34%.

Vectors, EXT1 V1 2021 SPEC2 12 MC

Consider the vectors  `underset~a = x underset~i + underset~j, \ underset~b = underset~i - underset~j`  and  `underset~c = underset~i + x underset~j`.

Given that  `theta`  is the angle between  `underset~a`  and  `underset~b`,  and  `phi`  is the angle between  `underset~b`  and  `underset~c, cos(theta) cos (phi)`  is

  1. `(2(1 + x^2))/(1 - x^2)`
  2. `(sqrt2(1 - x^2))/(1 + x^2)`
  3. `-((x + 1)^2)/(2(1 + x^2))`
  4. `-((x - 1)^2)/(2(1 + x^2))`
Show Answers Only

`D`

Show Worked Solution

`underset~a = x underset~i – underset~j, \ underset~b = underset~i – underset~j, \ underset~c = underset~i + x underset~j`

`underset~a · underset~b = |underset~a||underset~b|costheta`

`costheta = (x – 1)/(sqrt(x^2 + 1)sqrt2)`

`cos phi = (underset~b · underset~c)/(|underset~b||underset~c|) = (1 – x)/(sqrt2 sqrt(1 + x^2))`

`costheta · cos phi` `= ((x – 1)(1 – x))/(2(1 + x^2))`
  `= -((x – 1)^2)/(2(1 + x^2))`

 
`=>\ D`

Vectors, EXT1 V1 2021 HSC 5 MC

For the two vectors  `overset->(OA)`  and  `overset->(OB)`  it is know that

`overset->(OA) · overset->(OB) < 0`

Which of the following statements MUST be true?

  1. Either, `overset->(OA)`  is negative and  `overset->(OB)`  is positive, or  `overset->(OA)`  is positive and  `overset->(OB)`  is negative.
  2. The angle between  `overset->(OA)`  and  `overset->(OB)`  is obtuse.
  3. The product  `|overset->(OA)||overset->(OB)|`  is negative.
  4. The points `O`, `A` and `B` are collinear.
Show Answers Only

`B`

Show Worked Solution

`overset->(OA) · overset->(OB) < 0`

`|overset->(OA)||overset->(OB)| cos theta` `< 0`
`cos theta` `< 0`

 
`text(If)\ \ cos theta < 0, theta\ \ text{is in 2nd quadrant (obtuse).}`

`=> B`

Vectors, EXT1 V1 EQ-Bank 27

Using vectors, calculate the acute angle between the line that passes through  `A(1, 3)`  and  `B(2,–6)`  and the line that passes through  `C(1, 5)`  and  `D(3,–2)`.

Give your answer correct to one decimal place.  (2 marks)

Show Answers Only

`9.6°`

Show Worked Solution

`underset~a = ((1),(3)), \ underset~b = ((2),(−6)), \ underset~c = ((1),(5)), \ underset~d = ((3),(−2))`

`overset(->)(AB)` `= underset~b – underset~a = ((2),(−6)) – ((1),(3)) = ((1),(−9))`
`overset(->)(CD)` `= underset~d – underset~c = ((3),(−2)) – ((1),(5)) = ((2),(−7))`

 

`costheta` `= (overset(->)(AB) · overset(->)(CD))/(|overset(->)(AB)| · |overset(->)(CD)|)`
  `= (2 + 63)/(sqrt82 · sqrt53)`
  `= 0.985…`

 

`:. theta` `=cos^(-1) 0.985…`
  `= 9.605…`
  `= 9.6°\ \ (text(to 1 d.p.))`

Vectors, EXT1 V1 EQ-Bank 5 MC

What is the angle between the vectors  `((2),(1))`  and  `((-4),(2))`?

A.   `cos^(-1)(0.06)`

B.   `cos^(-1)(–0.06)`

C.   `cos^(-1)(0.6)`

D.   `cos^(-1)(–0.6)`

Show Answers Only

`D`

Show Worked Solution
`cos theta` `=(underset~a * underset~b)/(|underset~a||underset~b|)`  
  `=(-8+2)/(sqrt(2^2+1^2) xx sqrt((-4)^2+2^2)`  
  `=(-6)/(sqrt5 sqrt20)`  
  `=-0.6`  
`:. theta` `= cos^(-1) (-0.6)`  

  
`=> D`

Vectors, EXT1 V1 EQ-Bank 29

If  `theta`  is the angle between  `underset~a = underset~i + 3j`  and  `underset~b = 3underset~i + underset~j`, then find the exact value of  `cos 2theta`.  (2 marks)

Show Answers Only

`−7/25`

Show Worked Solution

`underset~a = [(1),(3)],\ \ underset~b = [(3),(1)]`

`|underset~a| = sqrt(1^2 + 3^2) = sqrt10`

`|underset~b| = sqrt(3^2 + 1^2) = sqrt10`

`cos theta= (1 xx 3 + 3 xx 1)/(sqrt10 sqrt10)= 3/5`

`cos2theta` `= 2cos^2theta – 1`
  `= 2(3/5)^2 – 1`
  `= −7/25`

Vectors, EXT1 V1 EQ-Bank 31

Relative to a fixed origin, the points `A`, `B` and `C` are defined respectively by the position vectors  `underset~a = −underset~i - underset~j, \ underset~b = 3underset~i + 2underset~j`  and  `underset~c = −aunderset~i + 2underset~j`, where  `a`  is a real constant.

If the magnitude of angle `ABC`  is  `pi/3`, find `a`.  (3 marks)

Show Answers Only

`−3`

Show Worked Solution

`text(Angle between)\ overset(->)(BA)\ text(and)\ overset(->)(BC) = pi/3`

`overset(->)(BA)` `= overset(->)(OA) – overset(->)(OB)`
  `= [(−1),(−1)] – [(3),(2)] = [(−4),(−3)]`

 

`overset(->)(BC)` `= overset(->)(OC) – overset(->)(OB)`
  `= [(−a),(2)] – [(3),(2)] = [(−a−3),(0 )]`

 

`overset(->)(BA) · overset(->)(BC)` `= [(−4),(−3)] · [(−a −3),(0 )]`
  `= 4a + 12`

 
`overset(->)(BA) · overset(->)(BC) = |overset(->)(BA)| · |overset(->)(BC)|costheta`

`4a + 12` `= sqrt((−4)^2 + (−3)^2) · sqrt((-a-3)^2) · cos\ pi/3`
`4a + 12` `= 5(-a-3) · 1/2`
`4a + 12` `= -(5a)/2-15/2`
`(13a)/2` `= -39/2`
`:.a` `= -3`

Vectors, EXT1 V1 EQ-Bank 20

Consider the vector  `underset~a = underset~i + sqrt3underset~j`, where  `underset~i`  and  `underset~j`  are unit vectors in the positive direction of the `x` and `y` axes respectively.

  1. Find the unit vector in the direction of  `underset~a`.    (1 mark)

  2. Find the acute angle that  `underset~a`  makes with the positive direction of the `x`-axis.   (1 mark)

  3. The vector  `underset~b = m underset~i - 2underset~j`.

     

    Given that  `underset~b`  is perpendicular to  `underset~a`, find the value of  `underset~m`.   (1 mark)

Show Answers Only
  1. `1/2(underset~i + sqrt3underset~j)`
  2. `60°`
  3. `2sqrt3`
Show Worked Solution

i.   `underset~a = underset~i + sqrt3underset~j`

`|underset~a| = sqrt(1 + (sqrt(3))^2) = 2`

`overset^a = (underset~a)/(|underset~a|) = 1/2(underset~i + sqrt3underset~j)`

 

ii.   `text(Solution 1)`

`underset~a\ =>\ text(Position vector from)\ \ O\ \ text{to}\ \ (1, sqrt3)`

`tan theta` `=sqrt3`  
`:. theta` `=60°`  
     

`text(Solution 2)`

`text(Angle with)\ xtext(-axis = angle with)\ \ underset~b = underset~i`

`underset~a · underset~i = 1 xx 1 = 1`

`underset~a · underset~i` `= |underset~a||underset~i|costheta`
`1` `= 2 xx 1 xx costheta`
`costheta` `= 1/2`
`:. theta` `= 60°`

 

iii.   `underset~b = m underset~i – 2underset~j`

`underset~a · underset~b = [(1),(sqrt3)] · [(m),(−2)] = m – 2sqrt3`

`text(S)text(ince)\ underset~a ⊥ underset~b:`

`m – 2sqrt3` `= 0`
`m` `= 2sqrt3`

Vectors, EXT1 V1 EQ-Bank 23

Points  `A`  and  `B`  have position vectors  `underset~a = 2underset~i - 2underset~j`  and  `underset~b =  4i + sqrt2 underset~j` respectively.

Find the angle between  `underset~a`  and  `underset~b`  to the nearest minute.  (2 marks)

Show Answers Only

`64°28′`

Show Worked Solution

`underset~a = 2underset~i – 2underset~j \ => \ |underset~a| = sqrt(2^2 + 2^2) = sqrt8`

`underset~b = 4 i + sqrt2 j \ => \ |underset~b| = sqrt(4^2 + (sqrt2)^2) = sqrt18`

`underset~a · underset~b = |underset~a||underset~b|costheta`

`costheta` `= (underset~a · underset~b)/(|underset~a||underset~b|)`
  `= (8 – 2sqrt2)/(sqrt8sqrt18)`
  `= (8 – 2sqrt2)/12`
`:. theta` `=64.471…`
  `=64°28′\ \ \ text{(nearest minute)}`

Vectors, EXT1 V1 EQ-Bank 32

Consider the vectors given by  `underset ~a = m underset ~i + underset ~j`  and  `underset ~b = underset ~i + m underset ~j`, where  `m in R`.

Find the value(s) of  `m`  if the acute angle between  `underset ~a`  and  `underset ~b`  is 30°.   (2 marks)

Show Answers Only

`sqrt 3, 1/sqrt 3`

Show Worked Solution
`underset ~a *underset ~b` `= m xx 1 + 1 xxm`
  `=2m`
   
`underset ~a *underset ~b` `= |underset ~a||underset ~b| cos 30^@`
  `= sqrt(m^2 + 1) *sqrt(1 + m^2) *cos 30^@`
  `= {(m^2 + 1) sqrt 3}/2`

 

`{(m^2 + 1) sqrt 3}/2` `=2m`  
`m^2 sqrt 3 + sqrt 3` `=4m`  
`m^2 sqrt 3-4m + sqrt 3` `=0`  
`(sqrt 3 m)^2-4(sqrt 3 m) + 3` `=0`  
`(sqrt 3 m)^2-4(sqrt 3 m) + 2^2-1` `=0`  
`(sqrt 3 m-2)^2-1` `=0`  
`sqrt 3 m-2` `= +-1`  
`sqrt 3 m` `= 2 +- 1`  

 
`:. m = (2 +- 1)/sqrt 3 = 3/sqrt 3 or 1/sqrt 3`

`= sqrt 3, 1/sqrt 3`

Vectors, EXT1 V1 2018 SPEC2 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?

A.   `underset ~a\ text(is parallel to)\ underset ~b`

B.   `|underset ~a| = |underset ~b|`

C.   `underset ~a = underset ~b`

D.   `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`