Advertisements
Advertisements
प्रश्न
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 ______.
विकल्प
`(3vec"a" + 2vec"b")/3`
`vec"a"`
`(5vec"a" - vec"b")/3`
`(4vec"a" + vec"b")/3`
उत्तर
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 `(4vec"a" + vec"b")/3`.
Explanation:
Applying section formula the position vector of the required point is
`(2(vec"a" + vec"b") + 1(2vec"a" - vec"b"))/(2 + 1) = (4vec"a" + vec"b")/3`
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
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 P, Q, R, S with position vectors \[\vec{p}\], \[\vec{q}\], \[\vec{r}\], \[\vec{s}\] respectively such that 5 \[\vec{p}\] − 2 \[\vec{q}\] + 6 \[\vec{r}\] − 9 \[\vec{s}\] \[\vec{0}\], are coplanar. Also, find the position vector of the point of intersection of the line segments PR and QS.
Show that the line segments joining the mid-points of opposite sides of a quadrilateral bisects each other.
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 the position vectors of the points A (3, 4), B (5, −6) and C (4, −1) are \[\vec{a,}\] \[\vec{b,}\] \[\vec{c}\] respectively, compute \[\vec{a} + 2 \vec{b} - 3 \vec{c}\].
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 the vertices of a triangle are the points with position vectors \[a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} , b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} , c_1 \hat{i} + c_2 \hat{j} + c_3 \hat{k} ,\]
what are the vectors determined by its sides? Find the length of these vectors.
Find the position vector of a point R which divides the line segment joining points \[P \left( \hat{i} + 2 \hat{j} + \hat{k} \right) \text{ and Q }\left( - \hat{i} + \hat{j} + \hat{k} \right)\] internally 2:1.
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
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.
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 four points having position vectors
\[6 \hat{i} - 7 \hat{j} , 16 \hat{i} - 19 \hat{j} - 4 \hat{k} , 3 \hat{j} - 6 \hat{k} , 2 \hat{i} - 5 \hat{j} + 10 \hat{k}\] are coplanar.
Show that the four points A, B, C and D with position vectors \[\vec{a}\], \[\vec{b}\], \[\vec{c}\], \[\vec{d}\] respectively are coplanar if and only if \[3 \vec{a} - 2 \vec{b} + \vec{c} - 2 \vec{d} = \vec{0} .\]
If D is the mid-point of side BC of a triangle ABC such that \[\overrightarrow{AB} + \overrightarrow{AC} = \lambda \overrightarrow{AD} ,\] write the value of λ.
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 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
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 `2vec"a" - 3vec"b"` and `vec"a" + vec"b"` in the ratio 3:1 is ______.
