Advertisements
Advertisements
प्रश्न
Show that the points A (1, −2, −8), B (5, 0, −2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.
Advertisements
उत्तर
Given points \[A\left( 1, - 2, - 8 \right), B\left( 5, 0, - 2 \right), C\left( 11, 3, 7 \right)\]
Therefore,
\[\overrightarrow{AB} = 5 \hat{i} + 0 \hat{j} - 2 \hat{k} - \hat{i} + 2 \hat{j} + 8 \hat{k} = 4 \hat{i} + 2 \hat{j} + 6 \hat{k}\]
\[\overrightarrow{BC} = 11 \hat{i} + 3 \hat{j} + 7 \hat{k} - 5 \hat{i} + 2 \hat{k} = 6 \hat{i} + 3 \hat{j} + 9 \hat{k}\]
and, \[\overrightarrow{AC} = 11 \hat{i} + 3 \hat{j} + 7 \hat{k} - \hat{i} + 2 \hat{j} + 8 \hat{k} = 10 \hat{i} + 5 \hat{j} + 15 \hat{k}\]
Clearly, \[\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}\]
Hence A, B, C are collinear.
Suppose B divides in the ratio AC in the ratio \[\lambda: 1\]. Then the position vector B is \[\left( \frac{11\lambda + 1}{\lambda + 1} \right) \hat{i} + \left( \frac{3\lambda - 2}{\lambda + 1} \right) \hat{j} + \left( \frac{7\lambda - 8}{\lambda + 1} \right) \hat{k}\]
But the position vector of B is \[5 \hat{i} + 0 \hat{j} - 2 \hat{k} .\]
\[\frac{11\lambda + 1}{\lambda + 1} = 5, \frac{3\lambda - 2}{\lambda + 1} = 0 , \frac{7\lambda - 8}{\lambda + 1} = - 2\]
\[ \Rightarrow 11\lambda + 1 = 5\lambda + 5, 3\lambda - 2 = 0, 7\lambda - 8 = - 2\lambda - 2\]
\[ \Rightarrow 6\lambda = 4, 3\lambda = 2, 9\lambda = 6\]
\[ \Rightarrow \lambda = \frac{2}{3}, \lambda = \frac{2}{3}, \lambda = \frac{2}{3}\]
APPEARS IN
संबंधित प्रश्न
Classify the following measures as scalars and vectors:
(i) 15 kg
(ii) 20 kg weight
(iii) 45°
(iv) 10 meters south-east
(v) 50 m/sec2
Answer the following as true or false:
\[\vec{a}\] and \[\vec{a}\] are collinear.
Answer the following as true or false:
Two vectors having same magnitude are collinear.
If \[\vec{a}\] and \[\vec{b}\] are two non-collinear vectors having the same initial point. What are the vectors represented by \[\vec{a}\] + \[\vec{b}\] and \[\vec{a}\] − \[\vec{b}\].
Five forces \[\overrightarrow{AB,} \overrightarrow { AC,} \overrightarrow{ AD,}\overrightarrow{AE}\] and \[\overrightarrow{AF}\] act at the vertex of a regular hexagon ABCDEF. Prove that the resultant is 6 \[\overrightarrow{AO,}\] where O is the centre of hexagon.
If O is a point in space, ABC is a triangle and D, E, F are the mid-points of the sides BC, CA and AB respectively of the triangle, prove that \[\vec{OA} + \vec{OB} + \vec{OC} = \vec{OD} + \vec{OE} + \vec{OF}\]
If \[\vec{a}\], \[\vec{b}\], \[\vec{c}\] are non-zero, non-coplanar vectors, prove that the following vectors are coplanar:
(1) \[5 \vec{a} + 6 \vec{b} + 7 \vec{c,} 7 \vec{a} - 8 \vec{b} + 9 \vec{c}\text{ and }3 \vec{a} + 20 \vec{b} + 5 \vec{c}\]
Prove that the following vectors are coplanar:
\[\hat{i} + \hat{j} + \hat{k} , 2 \hat{i} + 3 \hat{j} - \hat{k}\text{ and }- \hat{i} - 2 \hat{j} + 2 \hat{k}\]
Prove that the following vectors are non-coplanar:
If \[\vec{a}\], \[\vec{a}\], \[\vec{c}\] are non-coplanar vectors, prove that the following vectors are non-coplanar: \[\vec{a} + 2 \vec{b} + 3 \vec{c} , 2 \vec{a} + \vec{b} + 3 \vec{c}\text{ and }\vec{a} + \vec{b} + \vec{c}\]
Show that the vectors \[\vec{a,} \vec{b,} \vec{c}\] given by \[\vec{a} = \hat{i} + 2 \hat{j} + 3 \hat{k} , \vec{b} = 2 \hat{i} + \hat{j} + 3 \hat{k}\text{ and }\vec{c} = \hat{i} + \hat{j} + \hat{k}\] are non coplanar.
Express vector \[\vec{d} = 2 \hat{i}-j- 3 \hat{k} , \text{ and }\text { as a linear combination of the vectors } \vec{a,} \vec{b}\text{ and }\vec{c} .\]
If \[\vec{a} + \vec{b} + \vec{c} = \vec{0} , \left| \vec{a} \right| = 3, \left| \vec{b} \right| = 5, \left| \vec{c} \right| = 7,\] then the angle between \[\vec{a} \text{ and } \vec{b}\] is
The vector (cos α cos β) \[\hat{i}\] + (cos α sin β) \[\hat{j}\] + (sin α) \[\hat{k}\] is a
If the vectors `hati - 2xhatj + 3 yhatk and hati + 2xhatj - 3yhatk` are perpendicular, then the locus of (x, y) is ______.
What is the length of the longer diagonal of the parallelogram constructed on \[5 \vec{a} + 2 \vec{b} \text{ and } \vec{a} - 3 \vec{b}\] if it is given that \[\left| \vec{a} \right| = 2\sqrt{2}, \left| \vec{b} \right| = 3\] and the angle between \[\vec{a} \text{ and } \vec{b}\] is π/4?
If \[\vec{a} , \vec{b} , \vec{c}\] are any three mutually perpendicular vectors of equal magnitude a, then \[\left| \vec{a} + \vec{b} + \vec{c} \right|\] is equal to
If the vectors \[3 \hat{i} + \lambda \hat{j} + \hat{k} \text{ and } 2 \hat{i} - \hat{j} + 8 \hat{k}\] are perpendicular, then λ is equal to
The projection of the vector \[\hat{i} + \hat{j} + \hat{k}\] along the vector of \[\hat{j}\] is
The vectors \[2 \hat{i} + 3 \hat{j} - 4 \hat{k}\] and \[a \hat{i} + \hat{b} j + c \hat{k}\] are perpendicular if
If \[\left| \vec{a} \right| = \left| \vec{b} \right|, \text{ then } \left( \vec{a} + \vec{b} \right) \cdot \left( \vec{a} - \vec{b} \right) =\]
If \[\vec{a} \text{ and } \vec{b}\] are unit vectors, then the greatest value of \[\sqrt{3}\left| \vec{a} + \vec{b} \right| + \left| \vec{a} - \vec{b} \right|\]
If the angle between the vectors \[x \hat{i} + 3 \hat{j}- 7 \hat{k} \text{ and } x \hat{i} - x \hat{j} + 4 \hat{k}\] is acute, then x lies in the interval
If \[\vec{a} \text{ and } \vec{b}\] are two unit vectors inclined at an angle θ, such that \[\left| \vec{a} + \vec{b} \right| < 1,\] then
Let \[\vec{a} , \vec{b} , \vec{c}\] be three unit vectors, such that \[\left| \vec{a} + \vec{b} + \vec{c} \right|\] =1 and \[\vec{a}\] is perpendicular to \[\vec{b}\] If \[\vec{c}\] makes angles α and β with \[\vec{a} and \vec{b}\] respectively, then cos α + cos β =
If θ is an acute angle and the vector (sin θ) \[\text{i}\] + (cos θ) \[\hat{j}\] is perpendicular to the vector \[\hat{i} - \sqrt{3} \hat{j} ,\] then θ =
If \[\vec{a} \text{ and }\vec{b}\] be two unit vectors and θ the angle between them, then \[\vec{a} + \vec{b}\] is a unit vector if θ =
Which of the following quantities requires both magnitude (size) and direction for its complete description?
Two cars are moving at 50 km/h toward Mumbai from different cities. Are their velocity vectors equal? Why?
What is a position vector?
In the graphical representation of a vector, what does the arrow length represent?
