Advertisements
Advertisements
Question
Find the position vector of the foot of perpendicular and the perpendicular distance from the point P with position vector
`2hati+3hatj+4hatk` to the plane `vecr` . `(2hati+hatj+3hatk)−26=0` . Also find image of P in the plane.
Advertisements
Solution
Let M be the foot of the perpendicular drawn from the point P(2, 3, 4) in the plane
`(2hati+hatj+3hatk)−26=0 or 2x+y+3z-26=0`
Then, PM is the normal to the plane. So, the direction ratios of PM are proportional to 2, 1, 3.
Since PM passes through P(2, 3, 4) and has direction ratios proportional to 2, 1, 3, so the equation of PM is
`(x−2)/2=(y−3)/1=(z−4)/3=r (say)`
Let the coordinates of M be (2r + 2, r + 3, 3r + 4). Since M lies in the plane 2x + y + 3z − 26 = 0,so
2(2r+2)+r+3+3(3r+4)−26=0
⇒4r+4+r+3+9r+12−26=0
⇒14r−7=0
`⇒r=1/2`
Therefore, the coordinates of M are
`(2r+2, r+3, 3r+4)=(2xx1/2+2, 1/2+3, 3xx1/2+4)=(3, 7/2, 11/2)`
Thus, the position vector of the foot of perpendicular are `3hati+72hatj+112hatk.`
Now,
Length of the perpendicular from P on to the given plane
`=∣(2xx2+1xx3+3xx4−26)/sqrt(4+1+9)∣`
`=7/sqrt14`
`=sqrt(7/2) units`
Let `Q(x_1, y_1, z_1)`
be the image of point P in the given plane.
Then, the coordinates of M are ` ((x_1+2)/2, (y_1+3)/2, (z_1+4)/2)`
But, the coordinates of M are `(3, 7/2, 11/2)`
`therefore ((x_1+2)/2, (y_1+3)/2, (z_1+4)/2)=(3, 7/2, 11/2)`
`⇒(x_1+2)/2=3, (y_1+3)/2=7/2, (z_1+4)/2=11/2`
`⇒x_1=4, y_1=4, z_1=7`
Thus, the coordinates of the image of the point P in the given plane are (4, 4, 7).
APPEARS IN
RELATED QUESTIONS
Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are -2, 1, -1, and -3, -4, 1.
If `bara, barb, bar c` are the position vectors of the points A, B, C respectively and ` 2bara + 3barb - 5barc = 0` , then find the ratio in which the point C divides line segment AB.
Find the value of 'p' for which the vectors `3hati+2hatj+9hatk and hati-2phatj+3hatk` are parallel
Represent graphically a displacement of 40 km, 30° east of north.
`veca and -veca` are collinear.
Find the direction cosines of the vector `hati + 2hatj + 3hatk`.
Find the direction cosines of the vector joining the points A (1, 2, -3) and B (-1, -2, 1) directed from A to B.
Find the value of x for which `x(hati + hatj + hatk)` is a unit vector.
Express \[\vec{AB}\] in terms of unit vectors \[\hat{i}\] and \[\hat{j}\], when the points are A (4, −1), B (1, 3)
Find \[\left| \vec{A} B \right|\] in each case.
Find the angle between the vectors \[\vec{a} \text{ and } \vec{b}\] \[\vec{a} = 2\hat{i} - \hat{j} + 2\hat{k} \text{ and } \vec{b} = 4\hat{i} + 4 \hat{j} - 2\hat{k}\]
Find the angle between the vectors \[\vec{a} = 2 \hat{i} - 3 \hat{j} + \hat{k} \text{ and } \vec{b} = \hat{i} + \hat{j} - 2 \hat{k}\]
Find the angle between the vectors \[\vec{a} = \hat{i} + 2 \hat{j} - \hat{k} , \vec{b} = \hat{i} - \hat{j} + \hat{k}\]
Dot product of a vector with \[\hat{i} + \hat{j} - 3\hat{k} , \hat{i} + 3\hat{j} - 2 \hat{k} \text{ and } 2 \hat{i} + \hat{j} + 4 \hat{k}\] are 0, 5 and 8 respectively. Find the vector.
If \[\hat{ a } \text{ and } \hat{b }\] are unit vectors inclined at an angle θ, prove that
\[\tan\frac{\theta}{2} = \frac{\left| \hat{a} -\hat{b} \right|}{\left| \hat{a} + \hat{b} \right|}\]
If \[\vec{a,} \vec{b,} \vec{c}\] are three mutually perpendicular unit vectors, then prove that \[\left| \vec{a} + \vec{b} + \vec{c} \right| = \sqrt{3}\]
If \[\left| \vec{a} + \vec{b} \right| = 60, \left| \vec{a} - \vec{b} \right| = 40 \text{ and } \left| \vec{b} \right| = 46, \text{ find } \left| \vec{a} \right|\]
Show that the vector \[\hat{i} + \hat{j} + \hat{k}\] is equally inclined to the coordinate axes.
Show that the vectors \[\vec{a} = \frac{1}{7}\left( 2 \hat{i} + 3 \hat{j} + 6 \hat{k} \right), \vec{b} = \frac{1}{7}\left( 3\hat{i} - 6 {j} + 2 \hat{k} \right), \vec{c} = \frac{1}{7}\left( 6 \hat{i} + 2 \hat{j} - 3 {k} \right)\] are mutually perpendicular unit vectors.
Find the magnitude of two vectors \[\vec{a} \text{ and } \vec{b}\] that are of the same magnitude, are inclined at 60° and whose scalar product is 1/2.
If A, B and C have position vectors (0, 1, 1), (3, 1, 5) and (0, 3, 3) respectively, show that ∆ ABC is right-angled at C.
Show that the vectors \[2 \hat{i} - 3 \hat{j} + 4 \hat{k}\text{ and }- 4 \hat{i} + 6 \hat{j} - 8 \hat{k}\] are collinear.
if `hat"i" + hat"j" + hat"k", 2hat"i" + 5hat"j", 3hat"i" + 2 hat"j" - 3hat"k" and hat"i" - 6hat"j" - hat"k"` respectively are the position vectors A, B, C and D, then find the angle between the straight lines AB and CD. Find whether `vec"AB" and vec"CD"` are collinear or not.
Position vector of a point P is a vector whose initial point is origin.
Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are `hati + 2hatj - hatk` and `-hati + hatj + hatk` respectively, internally the ratio 2:1.
Which physical quantity has magnitude as well as direction?
In the vector \[\vec{AB}\], what are A and B respectively?
Which notation can be used for the magnitude of vector \[\vec{AB}\]?
Which expression gives the magnitude of the position vector \[\vec{OP}\] for \[P(x,y,z)\]?
