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
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.
Write the position vector of the point which divides the join of points with position vectors `3veca-2vecb and 2veca+3vecb` in the ratio 2 : 1.
Show that the vector `hati + hatj + hatk` is equally inclined to the axes OX, OY, and OZ.
Show that the points A, B and C with position vectors `veca = 3hati - 4hatj - 4hatk`, `vecb = 2hati - hatj + hatk` and `vecc = hati - 3hatj - 5hatk`, respectively form the vertices of a right angled triangle.
Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of the x-axis.
Find a vector of magnitude 4 units which is parallel to the vector \[\sqrt{3} \hat{i} + \hat{j}\]
ABCD is a parallelogram. If the coordinates of A, B, C are (−2, −1), (3, 0) and (1, −2) respectively, find the coordinates of D.
Find the angle between the vectors \[\vec{a} \text{ and } \vec{b}\] \[\vec{a} = 3\hat{i} - 2\hat{j} - 6\hat{k} \text{ and } \vec{b} = 4 \hat{i} - \hat{j} + 8 \hat{k}\]
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 angles which the vector \[\vec{a} = \hat{i} -\hat {j} + \sqrt{2} \hat{k}\] makes with the coordinate axes.
The adjacent sides of a parallelogram are represented by the vectors \[\vec{a} = \hat{i} + \hat{j} - \hat{k}\text{ and }\vec{b} = - 2 \hat{i} + \hat{j} + 2 \hat{k} .\]
Find unit vectors parallel to the diagonals of the parallelogram.
If \[\vec{a,} \vec{b,} \vec{c}\] are three mutually perpendicular unit vectors, then prove that \[\left| \vec{a} + \vec{b} + \vec{c} \right| = \sqrt{3}\]
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.
If \[\vec{p} = 5 \hat{i} + \lambda \hat{j} - 3 \hat{k} \text{ and } \vec{q} = \hat{i} + 3 \hat{j} - 5 \hat{k} ,\] then find the value of λ, so that \[\vec{p} + \vec{q}\] and \[\vec{p} - \vec{q}\] are perpendicular vectors.
If either \[\vec{a} = \vec{0} \text{ or } \vec{b} = \vec{0}\] then \[\vec{a} \cdot \vec{b} = 0 .\] But the converse need not be true. Justify your answer with an example.
If \[\vec{a} = 2 \hat{i} + 2 \hat{j} + 3 \hat{k} , \vec{b} = - \hat{i} + 2 \hat{j} + \hat{k} \text{ and } \vec{c} = 3 \hat{i} + \hat{j}\] \[\vec{a} + \lambda \vec{b}\] is perpendicular to \[\vec{c}\] then find the value of λ.
Find the angles of a triangle whose vertices are A (0, −1, −2), B (3, 1, 4) and C (5, 7, 1).
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.
Show that the points \[A \left( 2 \hat{i} - \hat{j} + \hat{k} \right), B \left( \hat{i} - 3 \hat{j} - 5 \hat{k} \right), C \left( 3 \hat{i} - 4 \hat{j} - 4 \hat{k} \right)\] are the vertices of a right angled triangle.
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.
If points A, B and C have position vectors `2hati, hatj` and `2hatk` respectively, then show that ΔABC is an isosceles triangle.
Geometrically, a vector is represented by which of the following?
What does \[\vec{AB}\] represent?
Which notation can be used for the magnitude of vector \[\vec{AB}\]?
In three-dimensional geometry, what is the vector drawn from \[O(0,0,0)\] to \[P(x,y,z)\] called?
Which expression gives the magnitude of the position vector \[\vec{OP}\] for \[P(x,y,z)\]?
