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1.

If the direction cosines of a variable line in two adjacent points be `l, M, n and l+deltal,m+deltam+n+deltan` the small angle `deltatheta`as between the two positions is given by

Answer» Clearly, we have
`l^(2)+m^(2)+n^(2)=1" "...(i)`
and`(l+deltal)^(2)+(m+deltam)^(2)+(n+deltan)^(2)=1" "...(ii)`
Substracting (i) from (ii), we get
`(l+deltal)^(2)+(m+deltam)^(2)+(n+deltan)^(2)-(l^(2)+m^(2)+n^(2))=0`
`rArr (deltal)^(2)+(deltam)^(2)+(deltan)^(2)=-2(l*deltal+m*deltam+n*deltan)" "...(iii)`
`therefore cosdeltatheta=l*(l+deltal)+m*(m+deltam)+n*(n+deltan)`
`=(l^(2)+m^(2)+n^(2))+(l*deltal+m*deltam+n*deltan)`
`=1-(1)/(2)[(deltal)^(2)+(deltam)^(2)+(deltan)^(2)]` [using (i) and (iii)].
`therefore(deltal)^(2)+(deltam)^(2)+(deltan)^(2)=2(1-cosdeltatheta)`
`=4sin^(2)"(deltatheta)/(2)=4*((deltatheta)/(2))^(2)`
`[therefore (deltatheta)/(2) " being small, " sin""(deltatheta)/(2)=(deltatheta)/(2)]`
`(deltatheta)^(2)`.
Hence, `(deltatheta)^(2)=(deltal)^(2)+(deltam)^(2)+(deltan)^(2)`.
Discussion
2.

If the line segment joining the points A(7, p, 2) and B(q, -2, 5) be parallel to the line segment joining the points C(2, -3, 5) and D(-6, -15, `11), find the value of p and q.

Answer» Correct Answer - `p=4,q=3`
`a_(1)=(q-7),b_(1)=(-2-p),c_(1)=(5-2)=3`.
`a_(2)=(-6-2)=-8,b_(2)=(-15+3)=-12,c_(2)=(11-5)=6`.
Since AB||CD, we have `(a_(1))/(a_(2))=(b_(1))/(b_(2))=(c_(1))/(c_(2))`.
`therefore (q-7)/(-8)=(-2-p)/(-12)=(3)/(6)=(1)/(2)`.Find the p and q.
Discussion
3.

Show that the straight lines whose direction cosinesare given by the equations `a l+b m+c n=0a n dul^2+z m^2=v n^2+w n^2=0`are parallel or perpendicular as`(a^2)/u+(b^2)/v+(c^2)/w=0ora^2(v+w)+b^2(w+u)+c^2(u+v)=0.`

Answer» The given equation are
`al+bm+cn=0" "...(i)`
`"ul"^(2)+vm^(2)+wn^(2)=0." "...(ii)`
Putting `l=(-(bm+cn))/(a)` from (i) in (ii), we get
`(u(bm+cn)^(2))/(a^(2))+vm^(2)+wn^(2)=0`
`rArr(b^(2)u+a^(2)v)m^(2)+2ubcmn+(c^(2)u+a^(2)w)n^(2)=0`
`rArr (b^(2)u+a^(2)v)((m)/(n))^(2)+2ubc((m)/(n))+(c^(2)u+a^(2)w)=0." "...(iii)`
Let `(m_(1))/(n_(1))and (m_(2))/(n_(2))` be the roots of (iii).
Then,`(m_(1))/(n_(1))*(m_(2))/(n_(2))=(c^(2)u+a^(2)w)/(b^(2)u+a^(2)v)`
`rArr(m_(1)m_(2))/(c^(2)u+a^(2))=(n_(1)n_(2))/(b^(2)u+a^(2)v)=(l_(1)l_(2))/(b^(2)w+c^(2)v)=k` [by symmetry].
`thereforel_(1)l_(2)+m_(1)m_(2)+n_(1)n_(2)=k(b^(2)w+c^(2)v+c^(2)u+a^2w+b^(2)u+a^(2)v)`.
The given lines are mutually perpendicular
`iffl_(1)l_(2)+m_(1)m_(2)+n_(1)n_(2)=0`
`iffa^(2)(v+w)+b^(2)(w+u)+c^(2)(u+v)=0`
For the given lines to be parallel, the direction cosines must be equal.
`therefore` the roots of (iii) must be equal.
`therefore 4u^(2)b^(2)c^(2)-4(b^(2)u+a^(2)v)(c^(2)u+a^(2)w)=0iff(a^(2))/(u)+(b^(2))/(v)+(c^(2))/(w)=0`.
Discussion
4.

Find the angle made by the following vector with the coordinates axes: `(hati+hatj+hatk)``(hatj-hatk)` `(hati-4hatj+8hatk)`

Answer» Correct Answer - (i) `cos^(-1)((1)/(sqrt3)),cos^(-1)((1)/(sqrt3)),cos^(-1)((1)/(sqrt3))`
(ii) `(pi)/(2),(pi)/(4),(3pi)/(4)`
(iii) `cos^(-1)""((1)/(9)),cos^(-1)((4)/(9)),cos^(-1)((8)/(9))`
(i) Let `veca=hati-hatj+hatk`.
`cosalpha=((hati-hatj+hatk)*hati)/(|hati-hatj+hatk||hati|)=(1)/(sqrt3),cosbeta=((hati-hatj+hatk)*hatj)/(|hati-hatj+hatk||hatj|)=(-1)/(sqrt3)`
and `cosgamma=((hati-hatj+hatk)*hatk)/(|hati-hatj+hatk||hatk|)=(1)/(sqrt3)`.
Discussion
5.

Find the coordinates of the foot of perpendiculardrawn from th point `A(1,8,4)`to the line joining the points `B(0,-1,3)a n dC(2-3,-1)dot`

Answer» Correct Answer - `((-5)/(3),(2)/(3),(19)/(3))`
Discussion
6.

Find the angle between two lines whose directionratios are proportional to `1,1,2a n d(sqrt(3)-1),(-sqrt(3)-1),4`.

Answer» Correct Answer - `(pi)/(3)`
`sqrt(1^(2)+1^(2)+2^(2))=sqrt6 and sqrt((sqrt3-1)^(2)+(-sqrt3-1)^(2)+4^(2))=sqrt24`.
`therefore costheta={(1)/(sqrt6)xx((sqrt3-1))/(sqrt24)+(1)/(sqrt6)xx((-sqrt3-1))/(24)+(2)/(sqrt6)xx(4)/(sqrt24)}`
`=((sqrt3-1))/(12)+((-sqrt3-1))/(12)+(8)/(12)=(6)/(12)=(1)/(2)`
`therefore theta = (pi)/(3)`.
Discussion
7.

Find the angle between the vectors `vecr_(1)=(3hati-2hatj+hatk) and hatr_(2)=(4hati+5hatj+7hatk)`.

Answer» Correct Answer - `cos^(-1)""((3)/(2sqrt35))`
`costheta=(vecr_(1)*vecr_(2))/(|vecr_(1)||vecr_(2)|)=((3hati-2hatj+hatk)*(4hati+5hatj+7hatk))/({sqrt(3^(2)+(-2)^(2)+1^(2))}{sqrt(4^(2)+5^(2)+7^(2))})=(3)/(2sqrt35)`.
Discussion
8.

Find the direction of a line whose direction ratio are `2, -6, 3`.

Answer» Here, `a=2,b=-6,c=3`.
`therefore " " sqrt(a^(2)+b^(2)+c^(2))=sqrt(2^(2)+(-6)^(2)+3^(2))`
`=sqrt(4+36+9)=sqrt49=7`.
Hence, the direction cosines of the given line ar `(2)/(7),(-6)/(7),(3)/(7)`.
Discussion