Advertisements
Advertisements
प्रश्न
Find \[\vec{a} \cdot \vec{b}\] when
\[\vec{a} =\hat{i} - 2\hat{j} + \hat{k}\text{ and } \vec{b} = 4 \hat{i} - 4\hat{j} + 7 \hat{k}\]
उत्तर
We have
\[ \vec{a} = \hat{i} - 2 \hat{j}+ \hat{k}\text{ and } \vec{b} = 4 \hat{i} - \hat{j} +7 \hat{k}\]
\[ \vec{a} . \vec{b} =\left( \hat{i}- 2 \hat{j} + \hat{k}\right) . \left( 4 \hat{[i} - \hat{j}+7 \hat{k}\right)\]
\[ = \left( 1 \right)\left( 4 \right) + \left( - 2 \right)\left( - 4 \right) + \left( 1 \right)\left( 7 \right)\]
\[ = 4 + 8 + 7\]
\[ = 19\]
APPEARS IN
संबंधित प्रश्न
If `veca and vecb` are two vectors such that `|veca+vecb|=|veca|,` then prove that vector `2veca+vecb` is perpendicular to vector `vecb`
Show that the vectors `veca, vecb` are coplanar if `veca+vecb, vecb+vecc ` are coplanar.
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`
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 the magnitude of each of two vectors `veca` and `vecb` having the same magnitude such that the angle between them is 60° and their scalar product is `9/2`
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
\[\vec{a} = \lambda \hat{i} + 2\hat{j} + \hat{k} \text{ and } \vec{b} = 5\hat{i} - 9 \hat{j} + 2\hat{k}\]
\[\text{ If } \vec{a} = \hat{i} - \hat{j} \text{ and } \vec{b} = - \hat{j} + 2\hat{k} , \text{find} \left( \vec{a} - 2 \vec{b} \right) \cdot \left( \vec{a} + \vec{b} \right) .\]
For any two vectors \[\vec{a} \text{ and } \vec{b}\] write when \[\left| \vec{a} + \vec{b} \right| = \left| \vec{a} \right| + \left| \vec{b} \right|\] holds.
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 \[\vec{a} \text{ and } \vec{b}\] 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 \[\hat{a} , \hat{b}\] are unit vectors such that \[\hat{a} + \hat{b}\] is a unit vector, write the value of \[\left| \hat{a} - \hat{b} \right| .\]
If \[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 5 \text{ and } \vec{a} . \vec{b} = 2, \text{ find } \left| \vec{a} - \vec{b} \right| .\]
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 a vector satisfying \[\vec{a} . \hat{i} = \vec{a} . \left( \hat{i} + \hat{j} \right) = \vec{a} . \left( \hat{i} + \hat{j} + \hat{k} \right) = 1 .\]
If \[\vec{a} \text{ and } \vec{b}\] are mutually perpendicular unit vectors, write the value of \[\left| \vec{a} + \vec{b} \right| .\]
If \[\vec{a} , \vec{b} \text{ and } \vec{c}\] are mutually perpendicular unit vectors, write the value of \[\left| \vec{a} + \vec{b} + \vec{c} \right| .\]
Find the angle between the vectors \[\vec{a} = \hat{i} - \hat{j} + \hat{k} \text{ and } \vec{b} = \hat{i} + \hat{j} - \hat{k} .\]
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 \[\vec{b} \text{ on } \vec{a}\]
For what value of λ are the vectors \[\vec{a} = 2 \text{i} + \lambda \hat{j} + \hat{k} \text{ and } \vec{b} = \hat{i} - 2 \hat{j} + 3 \hat{k}\] perpendicular to each other?
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 two unit vectors such that \[\vec{a} + \vec{b}\] is also a unit vector, then find the angle between \[\vec{a}\] and \[\vec{b}\]
Let `vec("a") = hat"i" + 2hat"j" - 3hat"k"` and `vec("b") = 3hat"i" -"j" +2hat("k")` be two vectors. Show that the vectors `(vec("a")+vec("b"))` and `(vec("a")-vec("b"))`are perpendicular to each other.
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 ______.
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 `veca, vecb, vecc` are three non-zero unequal vectors such that `veca.vecb = veca.vecc`, then find the angle between `veca` and `vecb - vecc`.
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.
If the two vectors `3hati + αhatj + hatk` and `2hati - hatj + 8hatk` are perpendicular to each other, then find the value of α.
