Advertisements
Advertisements
प्रश्न
Find the value of x such that the four-point with position vectors,
`"A"(3hat"i"+2hat"j"+hat"k"),"B" (4hat"i"+"x"hat"j"+5hat"k"),"c" (4hat"i"+2hat"j"-2hat"k")`and`"D"(6hat"i"+5hat"j"-hat"k")`are coplaner.
Advertisements
उत्तर
Let A, B, C, D be the given points. Then,
`vec("AB") = (4hat"i"+"x"hat"j"+5hat"k")-(3hat"i"+2hat"j"+hat"k")=hat"i"+("x"-2)hat"j"+4hat"k"`
`vec("AC") = (4hat"i"+2hat"j"-2hat"k")-(3hat"i"+2hat"j"+hat"k")=hat"i"+0hat"j"-3hat"k"`
`vec("AD") = (6hat"i"+2hat"j"-hat"k")-(3hat"i"+2hat"j"+hat"k")=3hat"i"+3hat"j"-2hat"k"`
The given points are coplanar if vectors `vec("AB") ,vec("AC") ,vec("AD")` are corplaner.
Therefore,
`[vec("AB") vec("AC") vec("AD")] = 0`
`⇒|(1,("x"-2),4),(1,0,-3),(3,3,-2)|=0`
⇒ 1 (0+9) - (x-2) (-2+9) + 4(3-0) = 0
⇒ 35 - 7x = 0
⇒ x =5
Hence, all the four points are coplanar for x =5.
APPEARS IN
संबंधित प्रश्न
The two vectors `hatj+hatk " and " 3hati-hatj+4hatk` represent the two sides AB and AC, respectively of a ∆ABC. Find the length of the median through A
If \[\vec{a,} \vec{b}\] are the position vectors of A, B respectively, find the position vector of a point C in AB produced such that AC = 3 AB and that a point D in BA produced such that BD = 2BA.
Show that the line segments joining the mid-points of opposite sides of a quadrilateral bisects each other.
Prove by vector method that the internal bisectors of the angles of a triangle are concurrent.
If the position vector of a point (−4, −3) be \[\vec{a,}\] find \[\left| \vec{a} \right|\]
If the position vector \[\vec{a}\] of a point (12, n) is such that \[\left| \vec{a} \right|\] = 13, find the value (s) of n.
Find the coordinates of the tip of the position vector which is equivalent to \[\vec{A} B\], where the coordinates of A and B are (−1, 3) and (−2, 1) respectively.
Show that the points 2 \[\hat{i}, - \hat{i}-4 \] \[\hat{j}\] and \[-\hat{i}+4\hat{j}\] form an isosceles triangle.
Find a unit vector in the direction of the resultant of the vectors
\[\hat{i} - \hat{j} + 3 \hat{k} , 2 \hat{i} + \hat{j} - 2 \hat{k} \text{ and }\hat{i} + 2 \hat{j} - 2 \hat{k} .\]
If \[\overrightarrow{PQ} = 3 \hat{i} + 2 \hat{j} - \hat{k}\] and the coordinates of P are (1, −1, 2), find the coordinates of Q.
Show that the points A, B, C with position vectors \[\vec{a} - 2 \vec{b} + 3 \vec{c} , 2 \vec{a} + 3 \vec{b} - 4 \vec{c}\] and \[- 7 \vec{b} + 10 \vec{c}\] are collinear.
Prove that the points having position vectors \[\hat{i} + 2 \hat{j} + 3 \hat{k} , 3 \hat{i} + 4 \hat{j} + 7 \hat{k} , - 3 \hat{i} - 2 \hat{i} - 5 \hat{k}\] are collinear.
If \[\vec{a,} \vec{b}\] are two non-collinear vectors prove that the points with position vectors \[\vec{a} + \vec{b,} \vec{a} - \vec{b}\text{ and }\vec{a} + \lambda \vec{b}\] are collinear for all real values of λ.
Show that the points whose position vectors are as given below are collinear:
\[2 \hat{i} + \hat{j} - \hat{k} , 3 \hat{i} - 2 \hat{j} + \hat{k} \text{ and }\hat{i} + 4 \hat{j} - 3 \hat{k}\]
Show that the points whose position vectors are as given below are collinear: \[3 \hat{i} - 2 \hat{j} + 4 \hat{k}, \hat{i} + \hat{j} + \hat{k}\text{ and }- \hat{i} + 4 \hat{j} - 2 \hat{k}\]
Define position vector of a point.
Find the image P' of the point P having position vector `hati+ 3hatj+ 4hatk` in the plane `vecr. (2hati - hatj + hatk) + 3 = 0 .` Hence find the length of PP'.
Find the position vector of the point which divides the join of points with position vectors `vec"a" + 3vec"b" and vec"a"- vec"b"` internally in the ratio 1 : 3.
Find the position vector of a point R which divides the line joining the two points P and Q with position vectors `vec"OP" = 2vec"a" + vec"b"` and `vec"OQ" = vec"a" - 2vec"b"`, respectively, in the ratio 1:2 externally
The position vector of the point which divides the join of points with position vectors `vec"a" + vec"b"` and 2`vec"a" - vec"b"` in the ratio 1:2 is ______.
The position vector of the point which divides the join of points `2vec"a" - 3vec"b"` and `vec"a" + vec"b"` in the ratio 3:1 is ______.
