Define Position Vector of a Point. - Mathematics

प्रश्न

Define position vector of a point.

बेरीज

उत्तर

A point O is fixed as origin in space (or plane) and P is any point, then $\vec{O P}$ is called a position vector of P with respect to O.

shaalaa.com

Position Vector of a Point Dividing a Line Segment in a Given Ratio

या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?

Q 3 Q 2 Q 4

पाठ 23: Algebra of Vectors - Very Short Answers [पृष्ठ ७५]

APPEARS IN

आरडी शर्मा Mathematics [English] Class 12

पाठ 23 Algebra of Vectors
Very Short Answers | Q 3 | पृष्ठ ७५

संबंधित प्रश्‍न

The two vectors $\hat{j} + \hat{k} and 3 \hat{i} - \hat{j} + 4 \hat{k}$ 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 $\vec{O P} = 2 \vec{a} + \vec{b}$ and $\vec{O Q} = \vec{a} - 2 \vec{b}$ , respectively in the ratio 1 : 2 internally and externally.

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 $\vec{d} = \frac{β \vec{b} + γ \vec{c}}{β + γ}, where β = | \vec{c} - \vec{a} | and, γ = | \vec{a} - \vec{b} |$
Hence, deduce that the incentre I has position vector
$\frac{α \vec{a} + β \vec{b} + γ \vec{c}}{α + β + γ}, where α = | \vec{b} - \vec{c} |$

If the position vector of a point (−4, −3) be $\vec{a,}$ find $| \vec{a} |$

If the position vector $\vec{a}$ of a point (12, n) is such that $| \vec{a} |$ = 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 $\vec{a}$ be the position vector whose tip is (5, −3), find the coordinates of a point B such that $\vec{A B} =$ $\vec{a}$ , 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 $λ \hat{i} +$ 3 $\hat{j}$ ,12 $\hat{i} + μ$ $\hat{j}$ 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 $λ$ and $μ$

Find a unit vector in the direction of the resultant of the vectors
$\hat{i} - \hat{j} + 3 \hat{k}, 2 \hat{i} + \hat{j} - 2 \hat{k} and \hat{i} + 2 \hat{j} - 2 \hat{k} .$

If $\vec{P Q} = 3 \hat{i} + 2 \hat{j} - \hat{k}$ 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:

$P (\hat{i} + 2 \hat{j} + \hat{k}) and Q (- \hat{i} + \hat{j} + \hat{k})$ externally

Find the position vector of the mid-point of the vector joining the points P (2, 3, 4) and Q(4, 1, −2).

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 $\vec{a} + \vec{b,} \vec{a} - \vec{b} and \vec{a} + λ \vec{b}$ 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: $3 \hat{i} - 2 \hat{j} + 4 \hat{k}, \hat{i} + \hat{j} + \hat{k} and - \hat{i} + 4 \hat{j} - 2 \hat{k}$

If D is the mid-point of side BC of a triangle ABC such that $\vec{A B} + \vec{A C} = λ \vec{A D},$ write the value of λ.

Find the image P' of the point P having position vector $\hat{i} + 3 \hat{j} + 4 \hat{k}$ in the plane $\vec{r} . (2 \hat{i} - \hat{j} + \hat{k}) + 3 = 0 .$ Hence find the length of PP'.

Find the value of x such that the four-point with position vectors,
$A (3 \hat{i} + 2 \hat{j} + \hat{k}), B (4 \hat{i} + x \hat{j} + 5 \hat{k}), c (4 \hat{i} + 2 \hat{j} - 2 \hat{k})$ and $D (6 \hat{i} + 5 \hat{j} - \hat{k})$ are coplaner.

Find the position vector of a point R which divides the line joining the two points P and Q with position vectors $\vec{OP} = 2 \vec{a} + \vec{b}$ and $\vec{OQ} = \vec{a} - 2 \vec{b}$ , 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 $\vec{OP} = 2 \vec{a} + \vec{b}$ and $\vec{OQ} = \vec{a} - 2 \vec{b}$ , respectively, in the ratio 1:2 externally

The position vector of the point which divides the join of points $2 \vec{a} - 3 \vec{b}$ and $\vec{a} + \vec{b}$ in the ratio 3:1 is ______.

Define Position Vector of a Point. - Mathematics

Advertisements

Advertisements

प्रश्न

उत्तर

APPEARS IN

संबंधित प्रश्‍न

Select a course

Define Position Vector of a Point. - Mathematics

Advertisements

Advertisements

प्रश्न

उत्तरShow Solution

APPEARS IN

संबंधित प्रश्‍न

Select a course

उत्तर