Advertisements
Advertisements
Question
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.
Solution
Let A, B, C be the points with position vectors
\[\hat{i} + 2 \hat{j} + 3 \hat{k} , 3 \hat{i} + 4 \hat{j} + 7 \hat{k} , - 3 \hat{i} - 2 \hat{j} - 5 \hat{k} .\]
Then,
\[\overrightarrow{AB} =\] Position vector of B - Position vector of A
\[= 3 \hat{i} + 4 \hat{j} + 7 \hat{k} - \hat{i} - 2 \hat{j} - 3 \hat{k} \]
\[ = 2 \hat{i} + 2 \hat{j} + 4 \hat{k}\]
\[\overrightarrow{BC} =\] Position vector of C - Position vector of B
\[= - 3 \hat{i} - 2 \hat{j} - 5 \hat{k} - 3 \hat{i} - 4 \hat{j} - 7 \hat{k} \]
\[ = - 6 \hat{i} - 6 \hat{j} - 12 \hat{k} \]
\[ = - 3\left( 2 \hat{i} + 2 \hat{j} + 4 \hat{k} \right)\]
∴ \[\overrightarrow{BC} = - 3 \overrightarrow{AB}\]
\[\text{ So, }\overrightarrow{AB} \text{ and }\overrightarrow{BC}\] are parallel vectors.
But B is a point common to them.
So,
\[\overrightarrow{AB}\] and \[\overrightarrow{BC}\] are collinear.
Hence,
A, B, C are collinear.
APPEARS IN
RELATED QUESTIONS
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
Find the position vector of a point R which divides the line joining the two points P and Q with position vectors \[\vec{OP} = 2 \vec{a} + \vec{b}\] and \[\vec{OQ} = \vec{a} - 2 \vec{b}\], respectively in the ratio 1 : 2 internally and externally.
Let \[\vec{a,} \vec{b,} \vec{c,} \vec{d}\] be the position vectors of the four distinct points A, B, C, D. If \[\vec{b} - \vec{a} = \vec{c} - \vec{d}\], then show that ABCD is a parallelogram.
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 four points A, B, C, D with position vectors \[\vec{a,} \vec{b,} \vec{c,} \vec{d}\] respectively such that \[3 \vec{a} - 2 \vec{b} + 5 \vec{c} - 6 \vec{d} = 0,\] are coplanar. Also, find the position vector of the point of intersection of the line segments AC and BD.
The vertices A, B, C of triangle ABC have respectively position vectors \[\vec{a}\], \[\vec{b}\], \[\vec{c}\] with respect to a given origin O. Show that the point D where the bisector of ∠ A meets BC has position vector \[\vec{d} = \frac{\beta \vec{b} + \gamma \vec{c}}{\beta + \gamma},\text{ where }\beta = \left| \vec{c} - \vec{a} \right| \text{ and, }\gamma = \left| \vec{a} - \vec{b} \right|\]
Hence, deduce that the incentre I has position vector
\[\frac{\alpha \vec{a} + \beta \vec{b} + \gamma \vec{c}}{\alpha + \beta + \gamma},\text{ where }\alpha = \left| \vec{b} - \vec{c} \right|\]
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.
If \[\vec{a}\] be the position vector whose tip is (5, −3), find the coordinates of a point B such that \[\overrightarrow{AB} =\] \[\vec{a}\], the coordinates of A being (4, −1).
Show that the points 2 \[\hat{i}, - \hat{i}-4 \] \[\hat{j}\] and \[-\hat{i}+4\hat{j}\] form an isosceles triangle.
The position vectors of points A, B and C are \[\lambda \hat{i} +\] 3 \[\hat{j}\],12\[\hat{i} + \mu\] \[\hat{j}\] and 11\[\hat{i} -\] 3 \[\hat{j}\] respectively. If C divides the line segment joining A and B in the ratio 3:1, find the values of \[\lambda\] and \[\mu\]
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.
Find the position vector of a point R which divides the line segment joining points:
\[P \left( \hat{i} + 2 \hat{j} + \hat{k}\right) \text { and } Q \left( - \hat{i} + \hat{j} + \hat{k} \right)\] externally
Find the position vector of the mid-point of the vector joining the points P (2, 3, 4) and Q(4, 1, −2).
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.
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.
If the points A(m, −1), B(2, 1) and C(4, 5) are collinear, find the value of m.
Show that the points whose position vectors are as given below are collinear:
\[2 \hat{i} + \hat{j} - \hat{k} , 3 \hat{i} - 2 \hat{j} + \hat{k} \text{ and }\hat{i} + 4 \hat{j} - 3 \hat{k}\]
If \[\vec{a}\], \[\vec{b}\], \[\vec{c}\] are position vectors of the points A, B and C respectively, write the value of \[\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{AC} .\]
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.
X and Y are two points with position vectors `3vec("a") + vec("b")` and `vec("a")-3vec("b")`respectively. Write the position vector of a point Z which divides the line segment XY in the ratio 2 : 1 externally.
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.
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 internally
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 ______.
