Mathematics Direction RATIOS AND DIRECTION COSINES

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Direction RATIOS AND DIRECTION COSINES

Direction cosines :

If `alpha, beta, gamma` are the angles which vector `vec(a) = a_1hat(i) + a_2hat(j) +a_3hat(k)` makes with positive direction of the `x, y, z ` axes respectively then `alpha , beta, gamma` are called direction angles and their cosines `cos alpha, cos beta , cos gamma` are called the
direction cosines of the vector and are generally denoted `l, m, n` respectively.

Thus ` l= cos alpha , m = cos beta , n = cos gamma`

`text(Note :)`

`(i)` If a line makes `alpha, beta , gamma` with positive direction of `x, y, z` axes respectively then direction cosines of line will
be `cos alpha , cos beta , cos gamma` or `- cos alpha , - cos beta , - cos gamma`.

`(ii)` A unit vector along the line whose direction cosines are `cos alpha, cos beta, cos gamma` can be written as

` (cos alpha) hat(i) + (cos beta) hat(j) +(cos gamma) hat(k) ` .

`(iii)` If a vector `vec(a) = a_1 hat(i) +a_2 hat(j)+a_3hat(k)` makes angles `alpha , beta , gamma` with positive direction of `x, y, z` axes respectively

then `cos alpha = (vec(a) *hat(i)) /(| vec(a)| | hat(i)|)= a_1/ | vec(a)|, cos beta = (vec(a)*hat(j))/ (|vec(a) | | hat(j)|) =a_2/|vec(a) |` and `cos gamma = (vec(a)*hat(k))/( |vec(a) | |hat(j) |) =a_3 /| vec(a)|`

`:.` `cos^2alpha + cos^2beta + cos^2gamma = (a_1^2+ a_2^2 +a_3^2) /|vec(a) |^2 => cos^2 alpha +cos^2 beta =cos^2 gamma =1`

Also note that `sin^2alpha + sin^2beta + sin^2gamma = 2`

`(iv)` Direction cosines of `x`-axis are `(1, 0, 0)` or `(- 1, 0, 0)`.
`quad quad ` Direction cosines of `y`-axis are `(0, 1 , 0)` or `(0, - 1 , 0)`.
`quadquad` Direction cosines of `z`-axis are `(0, 0, 1)` or `(0, 0, - 1)`.

If `alpha, beta, gamma` are the angles which vector `vec(a) = a_1hat(i) + a_2hat(j) +a_3hat(k)` makes with positive direction of the `x, y, z ` axes respectively then `alpha , beta, gamma` are called direction angles and their cosines `cos alpha, cos beta , cos gamma` are called the
direction cosines of the vector and are generally denoted `l, m, n` respectively.

Thus ` l= cos alpha , m = cos beta , n = cos gamma`

`text(Note :)`

`(i)` If a line makes `alpha, beta , gamma` with positive direction of `x, y, z` axes respectively then direction cosines of line will
be `cos alpha , cos beta , cos gamma` or `- cos alpha , - cos beta , - cos gamma`.

`(ii)` A unit vector along the line whose direction cosines are `cos alpha, cos beta, cos gamma` can be written as

` (cos alpha) hat(i) + (cos beta) hat(j) +(cos gamma) hat(k) ` .

`(iii)` If a vector `vec(a) = a_1 hat(i) +a_2 hat(j)+a_3hat(k)` makes angles `alpha , beta , gamma` with positive direction of `x, y, z` axes respectively

then `cos alpha = (vec(a) *hat(i)) /(| vec(a)| | hat(i)|)= a_1/ | vec(a)|, cos beta = (vec(a)*hat(j))/ (|vec(a) | | hat(j)|) =a_2/|vec(a) |` and `cos gamma = (vec(a)*hat(k))/( |vec(a) | |hat(j) |) =a_3 /| vec(a)|`

`:.` `cos^2alpha + cos^2beta + cos^2gamma = (a_1^2+ a_2^2 +a_3^2) /|vec(a) |^2 => cos^2 alpha +cos^2 beta =cos^2 gamma =1`

Also note that `sin^2alpha + sin^2beta + sin^2gamma = 2`

`(iv)` Direction cosines of `x`-axis are `(1, 0, 0)` or `(- 1, 0, 0)`.
`quad quad ` Direction cosines of `y`-axis are `(0, 1 , 0)` or `(0, - 1 , 0)`.
`quadquad` Direction cosines of `z`-axis are `(0, 0, 1)` or `(0, 0, - 1)`.

Direction ratios :

If `a, b, c` are three numbers proportional to the direction cosines `l , m, n` of a straight line, then `a, b, c` are
called its direction ratios. They are also called direction numbers or direction components.

Hence, we have ` l /a =m /b =n /c = lambda` (say) `=> l = a lambda , m = b lambda, n=c lambda `

`:.` `l^2 + m^2 +n^2 =1` `=> (a^2 +b^2 +c^2)lambda^2 =1 => lambda = pm 1/ (sqrt (a^2 +b^2 +c^2))`

`:.` `l = pm a/ (sqrt (a^2+b^2 +c^2)), m = pm b /(sqrt (a^2 +b^2 +c^2))` and `n = pm c/(sqrt (a^2 +b^2 +c^2))`

`text(Note: )`

`(i)` Direction ratios of a line is not unique but infinite in number but direction cosines will be for a line will be
only two.(`l , m, n` or `- l, - m,- n`)

`(ii)` A vector along the line with direction ratios `a, b, c` can be `ahat(i) + bhat(j) + chat(k)`.

`(iii)` Direction ratios a line joining two points `A` and `B` are proportional to `x_2 - x_1, y_2 - y_1, z_2 - z _1`.

`(iv)` `text(Projection of a Point on a Line: )`
Let `P` be a point and `AB` be a given line. Draw perpendicular `PQ` from `P` on `AB` which meets it at `Q`. This
point `Q` is called projection of `P` on the line `AB`.

`(v)` `text(Projection of a Line Segment Joining Two Points on a Line:)`

Projection of the line segment joining two points `P(x_1 y_1 z_1)` and `Q (x_2, y_2, z_2)` on another line whose
direction cosines are `l, m, n` is `AB = | l (x_2 - x_1) + m (y_2 - y_1) + n (_z2 - z_ 1) |.`

`text(Proof:)`

Vector `vecPQ = (x_2 - +x_ 1) hat(i) + (y_2 - y_1) hat(j) + (z_2 - z_1)hat(k)`

A unit vector along another line `hat (a )= l hat(i) + m hat(j) + n hat(k)` .

`:.` Projection `AB` = Projection of `vec(PQ)` on `hat (a) = |vec(PQ)* hat (a) |/|hat (a) |`

`= |l (x_2 -x_1) + m (y_2 -y_1) +n (z_2 -z_1) |`

If `a, b, c` are three numbers proportional to the direction cosines `l , m, n` of a straight line, then `a, b, c` are
called its direction ratios. They are also called direction numbers or direction components.

Hence, we have ` l /a =m /b =n /c = lambda` (say) `=> l = a lambda , m = b lambda, n=c lambda `

`:.` `l^2 + m^2 +n^2 =1` `=> (a^2 +b^2 +c^2)lambda^2 =1 => lambda = pm 1/ (sqrt (a^2 +b^2 +c^2))`

`:.` `l = pm a/ (sqrt (a^2+b^2 +c^2)), m = pm b /(sqrt (a^2 +b^2 +c^2))` and `n = pm c/(sqrt (a^2 +b^2 +c^2))`

`text(Note: )`

`(i)` Direction ratios of a line is not unique but infinite in number but direction cosines will be for a line will be
only two.(`l , m, n` or `- l, - m,- n`)

`(ii)` A vector along the line with direction ratios `a, b, c` can be `ahat(i) + bhat(j) + chat(k)`.

`(iii)` Direction ratios a line joining two points `A` and `B` are proportional to `x_2 - x_1, y_2 - y_1, z_2 - z _1`.

`(iv)` `text(Projection of a Point on a Line: )`
Let `P` be a point and `AB` be a given line. Draw perpendicular `PQ` from `P` on `AB` which meets it at `Q`. This
point `Q` is called projection of `P` on the line `AB`.

`(v)` `text(Projection of a Line Segment Joining Two Points on a Line:)`

Projection of the line segment joining two points `P(x_1 y_1 z_1)` and `Q (x_2, y_2, z_2)` on another line whose
direction cosines are `l, m, n` is `AB = | l (x_2 - x_1) + m (y_2 - y_1) + n (_z2 - z_ 1) |.`

`text(Proof:)`

Vector `vecPQ = (x_2 - +x_ 1) hat(i) + (y_2 - y_1) hat(j) + (z_2 - z_1)hat(k)`

A unit vector along another line `hat (a )= l hat(i) + m hat(j) + n hat(k)` .

`:.` Projection `AB` = Projection of `vec(PQ)` on `hat (a) = |vec(PQ)* hat (a) |/|hat (a) |`

`= |l (x_2 -x_1) + m (y_2 -y_1) +n (z_2 -z_1) |`

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