Advertisements
Advertisements
प्रश्न
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
विकल्प
\[\frac{\pi}{6}\]
\[\frac{2\pi}{3}\]
\[\frac{5\pi}{3}\]
\[\frac{\pi}{3}\]
Advertisements
उत्तर
\[\frac{\pi}{3}\]
\[\text{ Given }, \left| \vec{a} \right| = 3, \left| \vec{b} \right| = 5 \text{ and } \left| \vec{c} \right| = 7 . . . \left( i \right)\]
\[\text{ Let } \theta \text{ be the angle between } \vec{a} \text{ and } \vec{b} .\]
\[\text{ Given that }\]
\[ \vec{a} + \vec{b} + \vec{c} = 0\]
\[ \Rightarrow \vec{a} + \vec{b} = - \vec{c} \]
\[ \Rightarrow \left| \vec{a} + \vec{b} \right| = \left| - \vec{c} \right|^2 \]
\[ \Rightarrow \left| \vec{a} \right|^2 + \left| \vec{b} \right|^2 + 2 \vec{a} . \vec{b} = \left| \vec{c} \right|^2 \]
\[ \Rightarrow 2 \vec{a} . \vec{b} = \left| \vec{c} \right|^2 - \left| \vec{a} \right|^2 - \left| \vec{b} \right|^2 \]
\[ \Rightarrow 2 \vec{a} . \vec{b} = 7^2 - 3^2 - 5^2...................... \left[ \text{ Using } \left( i \right) \right]\]
\[ \Rightarrow 2 \vec{a} . \vec{b} = 15\]
\[ \Rightarrow 2 \left| \vec{a} \right| \left| \vec{b} \right| \cos \theta = 15\]
\[ \Rightarrow 2 \left( 3 \right) \left( 5 \right) \cos \theta = 15 ...................\left[ \text{ Using } \left( i \right) \right]\]
\[ \Rightarrow \cos \theta = \frac{1}{2}\]
\[ \therefore \theta = \frac{\pi}{3}\]
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
Classify the following as scalars and vector quantities:
(i) Time period
(ii) Distance
(iii) displacement
(iv) Force
(v) Work
(vi) Velocity
(vii) Acceleration
Answer the following as true or false:
Zero vector is unique.
Answer the following as true or false:
Two vectors having same magnitude are collinear.
If \[\vec{a}\] is a vector and m is a scalar such that m \[\vec{a}\] = \[\vec{0}\], then what are the alternatives for m and \[\vec{a}\] ?
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 \[\vec{OA} + \vec{OB} + \vec{OC} = \vec{OD} + \vec{OE} + \vec{OF}\]
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.
Prove that the following vectors are coplanar:
\[2 \hat{i} - \hat{j} + \hat{k} , \hat{i} - 3 \hat{j} - 5 \hat{k} \text{ and }3 \hat{i} - 4 \hat{j} - 4 \hat{k}\]
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: \[2 \vec{a} - \vec{b} + 3 \vec{c} , \vec{a} + \vec{b} - 2 \vec{c}\text{ and }\vec{a} + \vec{b} - 3 \vec{c}\]
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
Let \[\vec{a} \text{ and } \vec{b}\] be two unit vectors and α be the angle between them. Then, \[\vec{a} + \vec{b}\] is a unit vector if
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 ______.
The vector component of \[\vec{b}\] perpendicular to \[\vec{a}\] is
If \[\vec{a}\] is a non-zero vector of magnitude 'a' and λ is a non-zero scalar, then λ \[\vec{a}\] is a unit vector if
The values of x for which the angle between \[\vec{a} = 2 x^2 \hat{i} + 4x \hat{j} + \hat{k} , \vec{b} = 7 \hat{i} - 2 \hat{j} + x \hat{k}\] is obtuse and the angle between \[\vec{b}\] and the z-axis is acute and less than \[\frac{\pi}{6}\] are
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 inclined at an angle θ, then the value of \[\left| \vec{a} - \vec{b} \right|\]
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
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 =
The orthogonal projection of \[\vec{a} \text{ on } \vec{b}\] is
In Figure ABCD is a regular hexagon, which vectors are:
(i) Collinear
(ii) Equal
(iii) Coinitial
(iv) Collinear but not equal.
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?
