Advertisements
Advertisements
Question
Prove that the following vectors are non-coplanar:
Solution
Let if possible the given vectors are coplanar. Then one of the given vector is expressible in terms of the other two.
We have,
\[\hat{i} + 2 \hat{j} + 3 \hat{k} = x(2 \hat{i} + \hat{j} + 3 \hat{k} ) + y( \hat{i} + \hat{j} + \hat{k} ) . \]
\[ = \hat{i} (2x + y) + \hat{j} (x + y) + \hat{k} (3x + y) . \]
\[ \Rightarrow 2x + y = 1, x + y = 2, 3x + y = 3 . \]
By solving the first two equation, we get
\[ \Rightarrow x = - 1, y = 3 .\]
Clearly these values of x and y does not satisfy the third equation.
Hence the given vectors are non-coplanar.
APPEARS IN
RELATED QUESTIONS
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:
Two collinear vectors are always equal in magnitude.
Answer the following as true or false:
Two vectors having same magnitude are collinear.
Answer the following as true or false:
Two collinear vectors having the same magnitude are equal.
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 (3, 4), (−5, 16) and (5, 1) are collinear.
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.
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 non-coplanar:
If \[\vec{a}\], \[\vec{a}\], \[\vec{c}\] are non-coplanar vectors, prove that the following vectors are non-coplanar: \[2 \vec{a} - \vec{b} + 3 \vec{c} , \vec{a} + \vec{b} - 2 \vec{c}\text{ and }\vec{a} + \vec{b} - 3 \vec{c}\]
If \[\vec{a}\], \[\vec{a}\], \[\vec{c}\] are non-coplanar vectors, prove that the following vectors are non-coplanar: \[\vec{a} + 2 \vec{b} + 3 \vec{c} , 2 \vec{a} + \vec{b} + 3 \vec{c}\text{ and }\vec{a} + \vec{b} + \vec{c}\]
If \[\vec{a} \cdot \text{i} = \vec{a} \cdot \left( \hat{i} + \hat{j} \right) = \vec{a} \cdot \left( \hat{i} + \hat{j} + \hat{k} \right) = 1,\] then \[\vec{a} =\]
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
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
If \[\vec{a} \text{ and } \vec{b}\] are unit vectors, then which of the following values of \[\vec{a} . \vec{b}\] is not possible?
The vector component of \[\vec{b}\] perpendicular to \[\vec{a}\] 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 θ is the angle between two vectors `veca` and `vecb` then, `veca * vecb` ≥ 0, only when
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
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
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 =
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 =
In Figure ABCD is a regular hexagon, which vectors are:
(i) Collinear
(ii) Equal
(iii) Coinitial
(iv) Collinear but not equal.
