Advertisements
Advertisements
Question
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).
Advertisements
Solution
Let O be the origin and let \[P\left( 5, - 3 \right)\] be the tip of the position vector \[\vec{a}\]. Then, \[\vec{a} = \overrightarrow {OP} = 5 \hat{i}-3 \hat{j}.\] Let the coordinate of B be \[\left( x, y \right)\] and A has coordinates \[\left( 4, - 1 \right)\]
Therefore,
\[\overrightarrow{AB}\] = Position vector of B - Position vector of A
\[= \left( x \hat{i} + y \hat{j} \right) - \left( 4 \hat{i} - \hat{j} \right)\]
\[ = \left( x - 4 \right) \hat{i} + \left( y + 1 \right) \bar{j}\]
Now,
\[\overrightarrow {AB} = \vec{a} \]
\[ \Rightarrow \left( x - 4 \right) \hat{i} + \left( y + 1 \right) \hat{j} = 5 \hat{i} - 3 \hat{j} \]
\[ \Rightarrow x - 4 = 5\text{ and }y + 1 = - 3\]
\[ \Rightarrow x = 9\text{ and }y = - 4\]
Hence, the coordinates of B are \[\left( 9, - 4 \right)\].
APPEARS IN
RELATED QUESTIONS
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 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.
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 \[\vec{a} + 2 \vec{b} - 3 \vec{c}\].
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.
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.
If the points with position vectors \[10 \hat{i} + 3 \hat{j} , 12 \hat{i} - 5 \hat{j}\text{ and a }\hat{i} + 11 \hat{j}\] are collinear, find the value of a.
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:
\[2 \hat{i} + \hat{j} - \hat{k} , 3 \hat{i} - 2 \hat{j} + \hat{k} \text{ and }\hat{i} + 4 \hat{j} - 3 \hat{k}\]
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 \[3 \vec{a} - 2 \vec{b} + \vec{c} - 2 \vec{d} = \vec{0} .\]
Define position vector of a point.
If \[\vec{a}\], \[\vec{b}\], \[\vec{c}\] are position vectors of the points A, B and C respectively, write the value of \[\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{AC} .\]
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 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 ______.
Position vector of the mid-point of line segment AB is `3hati + 2hatj - 3hatk`. If position vector of the point A is `2hati + 3hatj - 4hatk`, then position vector of the point B is ______.
