Advertisements
Advertisements
प्रश्न
Prove that `(veca + vecb).(veca + vecb)` = `|veca|^2 + |vecb|^2` if and only if `veca . vecb` are perpendicular, given `veca != vec0, vecb != vec0.`
उत्तर
`(veca + vecb) xx (veca + vecb) = |veca|^2 + |vecb|^2`
`veca xx veca + veca xx vecb + vecb xx veca + vecb xx vecb = |veca|^2 + |vecb|^2`
`|veca|^2 + 2veca xx vecb + |b|^2 = |veca|^2 + |vecb|^2`
`2veca xx vecb = 0`
`veca xx vecb = 0`
`veca, vecb` are perpendicular.
APPEARS IN
संबंधित प्रश्न
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`.
If `vec a=7hati+hatj-4hatk and vecb=2hati+6hatj+3hatk` , then find the projection of `vec a and vecb`
Show that each of the given three vectors is a unit vector:
`1/7 (2hati + 3hatj + 6hatj), 1/7(3hati - 6hatj + 2hatk), 1/7(6hati + 2hatj - 3hatk)`
Also, show that they are mutually perpendicular to each other.
For what value of λ are the vectors \[\vec{a} \text{ and }\vec{b}\] perpendicular to each other if \[\vec{a} = \lambda \hat{i} + 2 \hat{j} + \hat{k} \text{ and } \vec{b} = 4\hat{i} - 9 \hat{j} + 2\hat{k}\]
For what value of λ are the vectors \[\vec{a} \text{ and } \vec{b}\] perpendicular to each other if
\[\vec{a} = \lambda \hat{i} + 3 \hat{j} + 2 \hat{k}\text { and } \vec{b} = \hat{i} - \hat{j} + 3 \hat{k}\]
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 \[\vec{a} \text{ and } \vec{b} .\]
\[\vec{a} \text{ and } \vec{b}\] 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 \[\vec{a} \text{ on } \vec{b}\]
If the vectors \[3 \hat{i} + m \hat{j} + \hat{k} \text{ and } 2 \hat{i} - \hat{j} - 8 \hat{k}\] are orthogonal, find m.
If the vectors \[3 \hat{i} - 2 \hat{j} - 4 \hat{k}\text{ and } 18 \hat{i} - 12 \hat{j} - m \hat{k}\] are parallel, find the value of m.
For any two vectors \[\vec{a} \text{ and } \vec{b}\] write when \[\left| \vec{a} + \vec{b} \right| = \left| \vec{a} - \vec{b} \right|\] holds.
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 \[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 5 \text{ and } \vec{a} . \vec{b} = 2, \text{ find } \left| \vec{a} - \vec{b} \right| .\]
Write the projections of \[\vec{r} = 3 \hat{i} - 4 \hat{j} + 12 \hat{k}\] on the coordinate axes.
Write the projection of \[\hat{i} + \hat{j} + \hat{k}\] along the vector \[\hat{j}\]
Write a vector satisfying \[\vec{a} . \hat{i} = \vec{a} . \left( \hat{i} + \hat{j} \right) = \vec{a} . \left( \hat{i} + \hat{j} + \hat{k} \right) = 1 .\]
Find the projection of \[\vec{a} \text{ on } \vec{b} \text{ if } \vec{a} \cdot \vec{b} = 8 \text{ and } \vec{b} = 2 \hat{i} + 6 \hat{j} + 3 \hat{k} .\]
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 .
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 projection of the vector \[\hat{i} + 3 \hat{j} + 7 \hat{k}\] on the vector \[2 \hat{i} - 3 \hat{j} + 6 \hat{k}\]
Write the projection of the vector \[7 \hat{i} + \hat{j} - 4 \hat{k}\] on the vector \[2 \hat{i} + 6 \hat{j}+ 3 \hat{k} .\]
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.
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 ______.
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 ______.
If `hata` and `hatb` are unit vectors, then prove that `|hata + hatb| = 2 cos theta/2`, where θ is the angle between them.
Let `veca, vecb, vecc` be three vectors of magnitudes 3, 4 and 5 respectively. If each one is petpendicular to the sum of the other two vectors, then `|veca + vecb + vecc|` =
If `θ` be the angle between any two vectors `veca` and `vecb`, then `|veca * vecb| = |veca xx vecb|`, when `θ` is equal to
If the two vectors `3hati + αhatj + hatk` and `2hati - hatj + 8hatk` are perpendicular to each other, then find the value of α.
