The vectors `underset~u=([a],[2])` and `underset~v=([a-7],[4a-1])` are perpendicular.
What are the possible values of `a`? (2 marks)
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The vectors `underset~u=([a],[2])` and `underset~v=([a-7],[4a-1])` are perpendicular.
What are the possible values of `a`? (2 marks)
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`a=1, -2`
`text{If}\ \ underset~u ⊥ underset~v:`
`([a],[2])*([a-7],[4a-1])` | `=0` | |
`a(a-7)+2(4a-1)` | `=0` | |
`a^2-7a+8a-2` | `=0` | |
`a^2+a-2` | `=0` | |
`(a+2)(a-1)` | `=0` |
`:.a=1\ \ text{or}\ \ -2`
For what values(s) of `a` are the vectors `((a),(−1))` and `((2a - 3),(2))` perpendicular? (3 marks)
`a = −1/2\ text(or)\ 2`
`((a),(−1)) · ((2a – 3),(2))` | `= 0` |
`a(2a – 3) + (−1) xx 2` | `= 0` |
`2a^2 – 3a – 2` | `= 0` |
`(2a + 1)(a – 2)` | `= 0` |
`:. a = −1/2\ \ text(or)\ \ 2`
Two vectors are given by `underset~a = 2underset~i + m underset~j` and `underset~b = −5underset~i + n underset~j` where `m, n > 0`.
If `|underset~a| = 3` and `underset~a` is perpendicular to `underset~b`, find the values of `m` and `n`. (2 marks)
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`2sqrt5`
`underset~a = [(2),(m)],\ \ underset~b = [(−5),(n)]`
`text(Using)\ |underset~a| = 3:`
`3` | `= sqrt(2^2 + m^2)` |
`m^2` | `= 5` |
`:.m` | `= sqrt5,\ \ \ (m > 0)` |
`text(S)text(ince)\ underset~a ⊥ underset~b:`
`a · b` | `= 0` |
`2xx −5 + mn` | `= 0` |
`sqrt5 n` | `= 10` |
`n` | `= 10/sqrt5` |
`= 2sqrt5` |
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.
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Given that `underset~b` is perpendicular to `underset~a`, find the value of `underset~m`. (1 mark)
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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` |
Consider the following vectors
`overset(->)(OA) = 2underset~i + 2underset~j,\ \ overset(->)(OB) = 3underset~i - underset~j,\ \ overset(->)(OC) = 5underset~i + 3underset~j`
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i. `text(Find)\ overset(->)(AB):`
`overset(->)(OA) = [(2),(2)],\ \ overset(->)(OB)[(3),(−1)]`
`overset(->)(AB)` | `= overset(->)(OB) – overset(->)(OA)` |
`= [(3),(−1)] – [(2),(2)]` | |
`= [(1),(−3)]` | |
`= underset~i – 3underset~j` |
ii. | `overset(->)(AC)` | `= overset(->)(OC) – overset(->)(OA)` |
`= [(5),(3)] – [(2),(2)]` | ||
`= [(3),(1)]` | ||
`= 3underset~i + underset~j` |
`overset(->)(AB) · overset(->)(AC)` | `= 1 xx 3 + −3 xx 1=0` |
`=> AB ⊥ AC`
`:. DeltaABC\ text(has a right angle at)\ A.`
iii. `overset(->)(BC)\ text(is the hypotenuse)`
`overset(->)(BC)` | `= overset(->)(OC) – overset(->)(OB)` |
`= [(5),(3)] – [(3),(−1)]` | |
`= [(2),(4)]` |
`|overset(->)(BC)|` | `=\ text(length of hypotenuse)` |
`= sqrt(2^2 + 4^2)` | |
`= sqrt(20)` | |
`= 2sqrt5` |
The vectors `underset~a = 2underset~i + m underset~j` and `underset~b = m^2underset~i - underset~j` are perpendicular for
A. `m = −2` and `m = 0`
B. `m = 2` and `m = 0`
C. `m = -1/2` and `m = 0`
D. `m = 1/2` and `m = 0`
`D`
`underset ~a ⊥ underset ~b\ \ =>\ \ underset ~a ⋅ underset ~b=0`
`underset ~a ⋅ underset ~b` | `= 2m^2 + m(-1)` |
`0` | `= 2m^2 – m` |
`0` | `= m(2m – 1)` |
`:. m = 0, quad m = 1/2`
`=> D`
Consider the vectors
`underset~a = 6underset~i + 2underset~j,\ \ underset~b = 2underset~i - m underset~j`
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i. | `2underset~a – 3underset~b` | `= 2[(6),(2)] – 3[(2),(−m)]` |
`= [(12),(4)] – [(6),(−3m)]` | ||
`= [(6),(4 + 3m)]` |
ii. `underset~a = [(6),(2)], \ \ underset~b = [(2),(−m)]`
`|underset~b|` | `= sqrt(4 + m^2)` |
`3sqrt2` | `= sqrt(4 + m^2)` |
`18` | `= 4 + m^2` |
`m^2` | `= 14` |
`m` | `= ±sqrt14` |
iii. `text(If)\ \ underset~a ⊥ underset~b \ => \ underset~a · underset~b = 0`
`6 xx 2 + 2 xx – m` | `= 0` |
`2m` | `= 12` |
`:. m` | `= 6` |
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`
`A`
`|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)`
`=>|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`