Advertisements
Advertisements
प्रश्न
Prove that, for any three vectors \[\vec{a} , \vec{b} , \vec{c}\] \[\left[ \vec{a} + \vec{b} , \vec{b} + \vec{c} , \vec{c} + \vec{a} \right] = 2 \left[ \vec{a} , \vec{b} , \vec{c} \right]\].
Advertisements
उत्तर
We have:
\[\left[ \vec{a} + \vec{b} , \vec{b} + \vec{c} , \vec{c} + \vec{a} \right]\]
\[ = \left( \vec{a} + \vec{b} \right) . \left[ \left( \vec{b} + \vec{c} \right) \times \left( \vec{c} + \vec{a} \right) \right]\]
\[ = \left( \vec{a} + \vec{b} \right) . \left[ \vec{b} \times \vec{c} + \vec{b} \times \vec{a} + \vec{c} \times \vec{c} + \vec{c} \times \vec{a} \right] (\text { By distributive law })\]
\[ = \left( \vec{a} + \vec{b} \right) . \left[ \vec{b} \times \vec{c} + \vec{b} \times \vec{a} + \vec{c} \times \vec{a} \right] ( \because \vec{c} \times \vec{c} = 0)\]
\[ = \vec{a} . \left( \vec{b} \times \vec{c} \right) + \vec{b} . \left( \vec{b} \times \vec{c} \right) + \vec{a} . \left( \vec{b} \times \vec{a} \right) + \vec{b .} \left( \vec{b} \times \vec{a} \right) + \vec{a} . \left( \vec{c} \times \vec{a} \right) + \vec{b .} \left( \vec{c} \times \vec{a} \right)\]
\[ = \left[ \vec{a} , \vec{b} , \vec{c} \right] + \left[ \vec{b} , \vec{b} , \vec{c} \right] + \left[ \vec{a} , \vec{b} , \vec{a} \right] + \left[ \vec{b} , \vec{b} , \vec{a} \right] + \left[ \vec{a} , \vec{c} , \vec{a} \right] + \left[ \vec{b} , \vec{c} , \vec{a} \right] \]
\[ = \left[ \vec{a} , \vec{b} , \vec{c} \right] + \left[ \vec{b} , \vec{c} , \vec{a} \right] ( \because \text { scalar triple product with two equal vectors is } 0) \]
\[ = \left[ \vec{a} , \vec{b} , \vec{c} \right] + \left[ \vec{a} , \vec{b} , \vec{c} \right] \left( \because \left[ \vec{b} , \vec{c} , \vec{a} \right] = \left[ \vec{a} , \vec{b} , \vec{c} \right] \right)\]
\[ = 2\left[ \vec{a} , \vec{b} , \vec{c} \right]\]
Hence,
\[\left[ \vec{a} + \vec{b} , \vec{b} + \vec{c} , \vec{c} + \vec{a} \right] = 2\left[ \vec{a} , \vec{b} , \vec{c} \right]\]
APPEARS IN
संबंधित प्रश्न
Find the projection of the vector `hati+3hatj+7hatk` on the vector `2hati-3hatj+6hatk`
If `vec a, vec b, vec c` are unit vectors such that `veca+vecb+vecc=0`, then write the value of `vec a.vecb+vecb.vecc+vecc.vec a`.
The scalar product of the vector `veca=hati+hatj+hatk` with a unit vector along the sum of vectors `vecb=2hati+4hatj−5hatk and vecc=λhati+2hatj+3hatk` is equal to one. Find the value of λ and hence, find the unit vector along `vecb +vecc`
Prove that `(veca + vecb).(veca + vecb)` = `|veca|^2 + |vecb|^2` if and only if `veca . vecb` are perpendicular, given `veca != vec0, vecb != vec0.`
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
\[\vec{a} = \hat{j} - \hat{k} \text{ and } \vec{b} = 2 \hat{i} + 3 \hat{j} - 2 \hat{k}\]
For what value of λ are the vectors \[\vec{a} \text{ and } \vec{b}\] perpendicular to each other if
\[\vec{a} = \lambda \hat{i} + 2\hat{j} + \hat{k} \text{ and } \vec{b} = 5\hat{i} - 9 \hat{j} + 2\hat{k}\]
What is the angle between vectors \[\vec{a} \text{ and } \vec{b}\] with magnitudes 2 and \[\sqrt{3}\] respectively? Given \[\vec{a} . \vec{b} = \sqrt{3} .\]
\[\vec{a} \text{ and } \vec{b}\] 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 \[\vec{b}\] is a unit vector such that\[\left( \vec{a} + \vec{b} \right) . \left( \vec{a} - \vec{b} \right) = 8, \text{ find } \left| \vec{a} \right| .\]
If \[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 5 \text{ and } \vec{a} . \vec{b} = 2, \text{ find } \left| \vec{a} - \vec{b} \right| .\]
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 value of θ ∈(0, π/2) for which vectors \[\vec{a} = \left( \sin \theta \right) \hat{i} + \left( \cos \theta \right) \hat{j} \text{ and } \vec{b} = \hat{i} - \sqrt{3} \hat{j} + 2 \hat{k}\] are perpendicular.
Write the projection of \[\hat{i} + \hat{j} + \hat{k}\] along the vector \[\hat{j}\]
Write a vector satisfying \[\vec{a} . \hat{i} = \vec{a} . \left( \hat{i} + \hat{j} \right) = \vec{a} . \left( \hat{i} + \hat{j} + \hat{k} \right) = 1 .\]
Write the value of p for which \[\vec{a} = 3 \hat{i} + 2 \hat{j} + 9 \hat{k} \text{ and } \vec{b} = \hat{i} + p \hat{j} + 3 \hat{k}\] are parallel vectors .
Write the angle between two vectors \[\vec{a} \text{ and } \vec{b}\] with magnitudes \[\sqrt{3}\] and 2 respectively if \[\vec{a} \cdot \vec{b} = \sqrt{6} .\]
Find λ when the projection of \[\vec{a} = \lambda \hat{i} + \hat{j} + 4 \hat{k} \text{ on } \vec{b} = 2 \hat{i} + 6 \hat{j} + 3 \hat{k}\] is 4 units.
Write the projection of \[\vec{b} + \vec{c} \text{ on } \vec{a} \text{ when } \vec{a} = 2 \hat{i} - 2 \hat{j} + \hat{k} , \vec{b} = \hat{i} + 2 \hat{j} - 2 \hat{k} \text{ and } \vec{c} = 2 \hat{i} - \hat{j} + 4 \hat{k} .\]
If the vectors \[\vec{a}\] and \[\vec{b}\] are such that \[\left| \vec{a} \right| = 3, \left| \vec{b} \right| = \frac{2}{3}\] and \[\vec{a} \times \vec{b}\] is a unit vector, then write the angle between \[\vec{a}\] and \[\vec{b}\]
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.
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.
The angle between two vectors `vec"a"` and `vec"b"` with magnitudes `sqrt(3)` and 4, respectively, and `vec"a" * vec"b" = 2sqrt(3)` is ______.
If two vectors `veca` and `vecb` are such that `|veca|` = 2, `|vecb|` = 3 and `veca.vecb` = 4, then `|veca - 2vecb|` is equal to ______.
