Advertisements
Advertisements
Question
Prove that the following vectors are coplanar:
\[\hat{i} + \hat{j} + \hat{k} , 2 \hat{i} + 3 \hat{j} - \hat{k}\text{ and }- \hat{i} - 2 \hat{j} + 2 \hat{k}\]
Advertisements
Solution
Given the vectors \[P\left( \hat{i} + \hat{j} + \hat{k} \right), Q\left( 2\hat{i} +3 \hat{j} - \hat{k} \right)\] and \[R\left( - \hat{i} - 2\hat{j} +2 \hat{k} \right)\]
We know the three vectors are coplanar if one of them is expressible as a linear combination of the other two.
Let, \[\hat{i} + \hat{j} + \hat{k} = x \left( 2 \hat{i} + 3 \hat{j} - \hat{k} \right) + y \left( - \hat{i} - 2 \hat{j} + 2 \hat{k} \right) . \]
\[ = \hat{i} \left( 2x - y \right) + \hat{j} \left( 3x - 2y \right) + \hat{k} \left( - x + 2y \right) .\]
\[\Rightarrow 2x - y = 1 , 3x - 2y = 1, - x + 2y = 1\] [Equating the coefficients of \[\hat{i} , \hat{j} , \hat{k}\] respectively]
Solving first two of these equation, we get \[x = 1 , y = 1\]. Clearly these two values satisfy the third equation.
Hence, the given vectors are 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 collinear vectors having the same magnitude are equal.
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 \[\vec{a}\] − \[\vec{b}\].
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 \[\vec{a}\], \[\vec{b}\], \[\vec{c}\] 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:
\[2 \hat{i} - \hat{j} + \hat{k} , \hat{i} - 3 \hat{j} - 5 \hat{k} \text{ and }3 \hat{i} - 4 \hat{j} - 4 \hat{k}\]
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: \[\vec{a} + 2 \vec{b} + 3 \vec{c} , 2 \vec{a} + \vec{b} + 3 \vec{c}\text{ and }\vec{a} + \vec{b} + \vec{c}\]
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 \[\vec{a} \cdot \text{i} = \vec{a} \cdot \left( \hat{i} + \hat{j} \right) = \vec{a} \cdot \left( \hat{i} + \hat{j} + \hat{k} \right) = 1,\] 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
The vector (cos α cos β) \[\hat{i}\] + (cos α sin β) \[\hat{j}\] + (sin α) \[\hat{k}\] is a
If \[\vec{a} \text{ and } \vec{b}\] are unit vectors, then which of the following values of \[\vec{a} . \vec{b}\] is not possible?
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
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 \[\vec{a}\] 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
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 \[\vec{a} \text{ and } \vec{b}\] 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
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 =
Which of the following quantities requires both magnitude (size) and direction for its complete description?
What does a negative vector (-\[\vec A\]) represent?
What is a position vector?
