Advertisements
Advertisements
प्रश्न
Prove that the following vectors are non-coplanar:
उत्तर
Let if possible the given vectors are coplanar. Then one of the given vector is expressible in terms of the other two.
We have,
\[3 \hat{i} + \hat{j} - \hat{k} = x(2 \hat{i} - \hat{j} + 7 \hat{k} ) + y(7 \hat{i} - \hat{j} + 23 \hat{k} ) . \]
\[ = \hat{i} (2x + 7y) + \hat{j} ( - x - y) + \hat{k} (7x + 23y) . \]
\[ \Rightarrow 2x + 7y = 3 , x + y = - 1, 7x + 23y = - 1 . \]
By solving the first two equations, we get
\[ \Rightarrow x = - 2, y = 1 .\]
Clearly these values of x and y does not satisfy the third equation.
Hence the given vectors are non-coplanar.
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:
\[\vec{a}\] and \[\vec{a}\] are collinear.
Answer the following as true or false:
Two collinear vectors are always equal in magnitude.
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 \[\vec{a}\] 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 the vectors \[\vec{a} = 2 \hat{i} - 3 \hat{j}\] and \[\vec{b} = - 6 \hat{i} + m \hat{j}\] are collinear, find the value of m.
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.
Using vectors show that the points A (−2, 3, 5), B (7, 0, −1) C (−3, −2, −5) and D (3, 4, 7) are such that AB and CD intersect at the point P (1, 2, 3).
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:
\[\hat{i} + \hat{j} + \hat{k} , 2 \hat{i} + 3 \hat{j} - \hat{k}\text{ and }- \hat{i} - 2 \hat{j} + 2 \hat{k}\]
If \[\vec{a}\], \[\vec{a}\], \[\vec{c}\] are non-coplanar vectors, prove that the following vectors are non-coplanar: \[\vec{a} + 2 \vec{b} + 3 \vec{c} , 2 \vec{a} + \vec{b} + 3 \vec{c}\text{ and }\vec{a} + \vec{b} + \vec{c}\]
The vectors \[\vec{a} \text{ and } \vec{b}\] satisfy the equations \[2 \vec{a} + \vec{b} = \vec{p} \text{ and } \vec{a} + 2 \vec{b} = \vec{q} , \text{ where } \vec{p} = \hat{i} + \hat{j} \text{ and } \vec{q} = \hat{i} - \hat{j} .\] the angle between \[\vec{a} \text{ and } \vec{b}\] then
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 \[\vec{a}\] is a non-zero vector of magnitude 'a' and λ is a non-zero scalar, then λ \[\vec{a}\] is a unit vector if
If θ is the angle between two vectors `veca` and `vecb` then, `veca * vecb` ≥ 0, only when
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 \[\vec{a} , \vec{b} , \vec{c}\] are any three mutually perpendicular vectors of equal magnitude a, then \[\left| \vec{a} + \vec{b} + \vec{c} \right|\] is equal to
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 inclined at an angle θ, then the value of \[\left| \vec{a} - \vec{b} \right|\]
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 =
The orthogonal projection of \[\vec{a} \text{ on } \vec{b}\] is
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 =
