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Given `underset~a = 4underset~i - 3underset~j` and `underset~b = 7underset~i - underset~j`, what is the magnitude of the projection of `underset~a` onto `underset~b`. Give your answer in simplest form. (3 marks)
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`(31sqrt2)/10`
`underset~a = [(4),(−3)],\ \ underset~b = [(7),(−1)]`
| `text(proj)_(underset~b) underset~a` | `= (28 + 3)/(49 + 1)(7underset~i – underset~j)` |
| `= 31/50(7underset~i – underset~j)` | |
| `= 217/50 underset~i – 31/50 underset~j` |
| `|\ text(proj)_(underset~b) underset~a\ |` | `= sqrt((217/50)^2 + (31/50)^2)` |
| `= sqrt((217^2 + 31^2))/50` | |
| `= sqrt(48\ 050)/50` | |
| `= (155sqrt2)/50` | |
| `= (31sqrt2)/10` |
Find the projection of `underset~a` onto `underset~b` given `underset~a = 2underset~i + underset~j` and `b = 3underset~i - 2underset~j`. (2 marks)
`12/13underset~i – 8/13underset~j`
`underset~a = [(2),(1)],\ \ underset~b = [(3),(−2)]`COMMENT: Many teachers recommend column vector notation to simplify calculations and minimise errors – we agree!
| `text(proj)_(underset~b) underset~a` | `= (underset~a · underset~b)/(underset~b · underset~b) xx underset~b` |
| `= (6 – 2)/(9 + 4)(3underset~i – 2underset~j)` | |
| `= 4/13(3underset~i – 2underset~j)` | |
| `= 12/13underset~i – 8/13underset~j` |
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)
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`64°28′`
`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)}` |
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)
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`sqrt 3, 1/sqrt 3`
| `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`
The projection of the force `underset~F = a underset~i + b underset~j`, where `a` and `b` are non-zero real constants, in the direction of the vector `underset~w = underset~i + underset~j`, is
`A`
`hatw= underset~w/sqrt(1+1)= (underset~i + underset~j)/sqrt2`
`underset~F*hat w = (a + b)/sqrt2`♦ Mean mark 49%.
`(underset~F*hat w)hatw= ((a + b)/sqrt2) underset~w/sqrt2= ((a + b)/2) underset~w`
`=> A`
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)`♦♦♦ 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`