Consider the vectors \(\underset{\sim}{r}\) and \(\underset{\sim}{s}\)  where  \(\abs{\underset{\sim}{ r }}=9\)  and  \(\underset{\sim}{ s }=2 \underset{\sim}{ i }-2 \underset{\sim}{ j }+\underset{\sim}{k}\).

If the vector resolute of \(\underset{\sim}{r}\) in the direction of \(\underset{\sim}{s}\) is equal to  \(-4 \underset{\sim}{i}+4 \underset{\sim}{j}-2 \underset{\sim}{k}\),  then the scalar resolute of \(\underset{\sim}{s}\) in the direction of \(\underset{\sim}{r}\) is equal to

1. \(-18\)
2. \(-2\)
3. \(2\)
4. \(3\)

The scalar resolute of vector  `underset~a`  in the direction of vector  `underset~b`  is  `-4`.

If  `underset~b = – sqrt3 underset~i`,  the vector resolute of  `underset~a`  in the direction of  `underset~b`  is

1. `-4underset~i`
2. `-3underset~i`
3. `1/sqrt3 underset~i`
4. `3underset~i`
5. `4underset~i`

The magnitude of the component of the force  `underset~F = underset~i + 6underset~j - 18underset~k`  that acts in the direction  `underset~d = 2underset~i - 3underset~j - 6underset~k`  is

1. `92/19`
2. `92/7`
3. `124/7`
4. `92/11`
5. `18/7`

The vector resolute of  `underset~a = 2underset~i - underset~j + 3underset~k`  that is perpendicular to  `underset~b = underset~i + underset~j - underset~k`  is

1. `−2/3(underset~i + underset~j - underset~k)`
2. `−2/3(2underset~i - underset~j + 3underset~k)`
3. `1/3(8underset~i - underset~j + 7underset~k)`
4. `underset~i - 2underset~j + 4underset~k`
5. `underset~i + underset~j + 2underset~k`

The vector resolute of  `underset~i + underset~j - underset~k`  in the direction of  `m underset~i + n underset~j + p underset~k`  is  `2underset~i - 3underset~j + underset~k`,  where  `m, n`  and  `p`  are real constants.

The values of  `m, n`  and  `p`  can be found by solving the equations.

1. `(m(m + n - p))/(m^2 + n^2 + p^2) = 2, \ (n(m + n - p))/(m^2 + n^2 + p^2) = −3 and (p(m + n - p))/(m^2 + n^2 + p^2) = 1`
2. `(m(m + n - p))/(m^2 + n^2 + p^2) = 1, \ (n(m + n - p))/(m^2 + n^2 + p^2) = 1 and (p(m + n - p))/(m^2 + n^2 + p^2) = −1`
3. `m + n - p = 6, \ m + n - p = −9 and m + n - p = −3`
4. `m + n - p = 3m, \ m + n - p = 3n and m + n - p = −3p`
5. `m + n - p = 2sqrt3, \ m + n - p = −3sqrt3 and m + n - p = sqrt3`

If  `underset ~u = 2 underset ~i - 2 underset ~j + underset ~k`  and  `underset ~v = 3 underset ~i - 6 underset ~j + 2 underset ~k`, the vector resolute of  `underset ~v`  in the direction of  `underset ~u`  is

A.   `20/49(3 underset ~i - 6 underset ~j + 2 underset ~k)`

B.   `20/3(2 underset ~i - 2underset ~j + underset ~k)`

C.   `20/7(3 underset ~i - 6 underset ~j + 2 underset ~k)`

D.   `20/9(2 underset ~i - 2 underset ~j + underset ~k)`

E.   `1/9(−2 underset ~i + 2 underset ~j - underset ~k)`

The component of the force  `underset~F = aunderset~i + bunderset~j`, where `a` and `b` are non-zero real constants, in the direction of the vector  `underset~w = underset~i + underset~j`, is

A.   `((a + b)/2)underset~w`

B.   `underset~F/(a + b)`

C.   `((a + b)/(a^2 + b^2))underset~F`

D.   `(a + b)underset~w`

E.   `((a + b)/sqrt2)underset~w` 

If  `underset ~u = 3 underset ~i + 6 underset ~j - 2 underset ~k`  and  `underset ~v = 2 underset ~i + 2 underset ~j - underset ~k`, then the vector resolute of  `underset ~u`  in the direction of  `underset ~v`  is

A.   `20/7 (3 underset ~i + 6 underset ~j - 2 underset ~k)`

B.   `20/9 (2 underset ~i + 2 underset ~j - underset ~k)`

C.   `20/49 (3 underset ~i + 6 underset ~j - 2 underset ~k)`

D.   `20/3 (2 underset ~i + 2 underset ~j - underset ~k)`

E.   `3/7 (3 underset ~i + 6 underset ~j - 2 underset ~k)`

Let  `underset ~a = 3 underset ~i - 2 underset ~j + m underset ~k`  and  `underset ~b = 2 underset ~i - underset ~j + 3 underset ~k`, where  `m in R`.

Find the value(s) of `m` such that the magnitude of the vector resolute of  `underset ~a` parallel to `underset ~b`  is equal to  `sqrt 14`.  (3 marks)

The scalar resolute of  `underset ~a = 3 underset ~i - 2 underset ~k`  in the direction of  `underset ~b = - underset ~i + 2 underset ~j + 3 underset ~k`  is

A.   `-(9 sqrt 13)/13`

B.   `-9/14(-underset ~i + 2 underset ~j + 3 underset ~k)`

C.   `-(9 sqrt 14)/14`

D.   `-9/13 (3 underset ~i - 2 underset ~k)`

E.   `- sqrt 14/2` 

Given the vectors  `underset~a = 3underset~i - 4underset~j + 12underset~k`  and  `underset~b = 2underset~i + 2underset~j - underset~k`, the vector resolute of  `underset~a` in the direction of  `underset~b`  is

1.  `−14/3`
2.  `−14/3(2underset~i - 2underset~j - underset~k)`
3.  `−14/9(2underset~i + 2underset~j - underset~k)`
4.  `−14/13`
5.  `−14/169(3underset~i - 4underset~j + 12underset~k)`