Advertisements
Advertisements
Question
Show that the vectors \[\vec{a,} \vec{b,} \vec{c}\] are coplanar if and only if \[\vec{a} + \vec{b}\], \[\vec{b} + \vec{c}\] and \[\vec{c} + \vec{a}\] are coplanar.
Advertisements
Solution
Consider,
\[\left[ \left( \vec{a} + \vec{b} \right) \left( \vec{b} + \vec{c} \right) \left( \vec{c} + \vec{a} \right) \right] = \left( \vec{a} + \vec{b} \right) . \left\{ \left( \vec{b} + \vec{c} \right) \times \left( \vec{c} + \vec{a} \right) \right\} \]
\[ = \left( \vec{a} + \vec{b} \right) . \left( \vec{b} \times \vec{c} + \vec{b} \times \vec{a} + \vec{c} \times \vec{c} + \vec{c} + \vec{a} \right)\]
\[ = \left( \vec{a} + \vec{b} \right) . \left( \vec{b} \times \vec{c} + \vec{b} \times \vec{a} + \vec{c} \times \vec{a} \right) ( \because \vec{c} \times \vec{c} = \vec{0} )\]
\[ = \vec{a} . \left( \vec{b} \times \vec{c} \right) + \vec{a} . \left( \vec{b} \times \vec{a} \right) + \vec{a} . \left( \vec{c} \times \vec{a} \right) + \vec{b} . \left( \vec{b} \times \vec{c} \right) + \vec{b} . \left( \vec{b} \times \vec{a} \right) + \vec{b} . \left( \vec{c} \times \vec{a} \right)\]
\[ = \vec{a} . \left( \vec{b} \times \vec{c} \right) + \vec{b} . \left( \vec{c} \times \vec{a} \right)\]
\[ = \left[ \vec{a} \vec{b} \vec{c} \right] + \left[ \vec{b} \vec{c} \vec{a} \right]\]
\[ = 2\left[ \vec{a} \vec{b} \vec{c} \right]\]
Now, we can see that
\[\left[ ( \vec{a} + \vec{b} ) ( \vec{b} + \vec{c} ) ( \vec{c} + \vec{a} ) \right] = 2\left[ \vec{a} \vec{b} \vec{c} \right]\]
Hence, the vectors
\[\vec{a} , \vec{b} , \vec{c}\] are coplanar if and only if \[\vec{a} + \vec{b} , \vec{b} + \vec{c} , \vec{c} + \vec{a}\]are coplanar.
APPEARS IN
RELATED QUESTIONS
Find the projection of the vector `hati+3hatj+7hatk` on the vector `2hati-3hatj+6hatk`
If `vec a=7hati+hatj-4hatk and vecb=2hati+6hatj+3hatk` , then find the projection of `vec a and vecb`
The scalar product of the vector `hati + hatj + hatk` with a unit vector along the sum of vectors `2hati + 4hatj - 5hatk` and `lambdahati + 2hatj + 3hatk` is equal to one. Find the value of `lambda`.
Find `lambda` if the scalar projection of `vec a = lambda hat i + hat j + 4 hat k` on `vec b = 2hati + 6hatj + 3hatk` is 4 units
Find \[\vec{a} \cdot \vec{b}\] when
\[\vec{a} = \hat{j} + 2 \hat{k} \text{ and } \vec{b} = 2 \hat{i} + \hat{k}\]
For what value of λ are the vectors \[\vec{a} \text{ and } \vec{b}\] perpendicular to each other if
\[\vec{a} = 2 \hat{i} + 3 \hat{j} + 4\hat{k} \text{ and } \vec{b} = 3 \hat{i} - 2 \hat{j} +\lambda \hat{k}\]
If \[\vec{a} \text{ and } \vec{b}\] are two vectors such that \[\left| \vec{a} \right| = 4, \left| \vec{b} \right| = 3 \text{ and } \vec{a} \cdot \vec{b} = 6\] find the angle between \[\vec{a} \text{ and } \vec{b} .\]
\[\text{ If } \vec{a} = \hat{i} - \hat{j} \text{ and } \vec{b} = - \hat{j} + 2\hat{k} , \text{find} \left( \vec{a} - 2 \vec{b} \right) \cdot \left( \vec{a} + \vec{b} \right) .\]
\[\vec{a} \text{ and } \vec{b}\] are two vectors such that \[\vec{a} . \vec{b} = 6, \left| \vec{a} \right| = 3 \text{ and } \left| \vec{b} \right| = 4 .\] Write the projection of \[\vec{a} \text{ on } \vec{b}\]
Find the cosine of the angle between the vectors \[4 \hat{i} - 3 \hat{j} + 3 \hat{k} \text{ and } 2 \hat{i} - \hat{j} - \hat{k} .\]
If \[\vec{a} \text{ and } \vec{b}\] are two vectors of the same magnitude inclined at an angle of 60° such that \[\vec{a} . \vec{b} = 8,\] write the value of their magnitude.
If \[\vec{a} . \vec{a} = 0 \text{ and } \vec{a} . \vec{b} = 0,\] what can you conclude about the vector \[\vec{b}\]
For any two non-zero vectors, write the value of \[\frac{\left| \vec{a} + \vec{b} \right|^2 + \left| \vec{a} - \vec{b} \right|^2}{\left| \vec{a} \right|^2 + \left| \vec{b} \right|^2} .\]
Write the component of \[\vec{b}\] along \[\vec{a}\]
Find the angle between the vectors \[\vec{a} = \hat{i} - \hat{j} + \hat{k} \text{ and } \vec{b} = \hat{i} + \hat{j} - \hat{k} .\]
For what value of λ are the vectors \[\vec{a} = 2 \hat{i} + \lambda \hat{j} + \hat{k} \text{ and } \vec{b} = \hat{i} - 2 \hat{j} + 3 \hat{k}\] perpendicular to each other?
Find the value of λ if the vectors \[2 \hat{i} + \lambda \hat{j} + 3 \hat{k} \text{ and } 3 \hat{i} + 2 \hat{j} - 4 \hat{k}\] are perpendicular to each other.
If \[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 3 \text{ and } \vec{a} \cdot \vec{b} = 3,\] find the projection of \[\vec{b} \text{ on } \vec{a}\]
Write the angle between two vectors \[\vec{a} \text{ and } \vec{b}\] with magnitudes \[\sqrt{3}\] and 2 respectively if \[\vec{a} \cdot \vec{b} = \sqrt{6} .\]
If \[\vec{a}\] and \[\vec{b}\] are unit vectors, then find the angle between \[\vec{a}\] and \[\vec{b}\] given that \[\left( \sqrt{3} \vec{a} - \vec{b} \right)\] is a unit vector.
Prove that, for any three vectors \[\vec{a} , \vec{b} , \vec{c}\] \[\left[ \vec{a} + \vec{b} , \vec{b} + \vec{c} , \vec{c} + \vec{a} \right] = 2 \left[ \vec{a} , \vec{b} , \vec{c} \right]\].
If \[\vec{a} \text{ and } \vec{b}\] are two non-collinear unit vectors such that \[\left| \vec{a} + \vec{b} \right| = \sqrt{3},\] find \[\left( 2 \vec{a} - 5 \vec{b} \right) \cdot \left( 3 \vec{a} + \vec{b} \right) .\]
The vectors `vec"a" = 3hat"i" - 2hat"j" + 2hat"k"` and `vec"b" = -hat"i" - 2hat"k"` are the adjacent sides of a parallelogram. The acute angle between its diagonals is ______.
Three vectors `veca, vecb` and `vecc` satisfy the condition `veca + vecb + vecc = vec0`. Evaluate the quantity μ = `veca.vecb + vecb.vecc + vecc.veca`, if `|veca|` = 3, `|vecb|` = 4 and `|vecc|` = 2.
