Advertisements
Advertisements
Question
For what value of λ are the vectors \[\vec{a} \text{ and } \vec{b}\] perpendicular to each other if
\[\vec{a} = \lambda \hat{i} + 2\hat{j} + \hat{k} \text{ and } \vec{b} = 5\hat{i} - 9 \hat{j} + 2\hat{k}\]
Advertisements
Solution
\[\text{ If the vectors } \vec{a} \text{ and } \vec{b} \text{ are perpendicular to each other, then }\]
\[ \vec{a} . \vec{b} = 0\]
\[ \Rightarrow \left( \lambda \hat{i} + 2 \hat{j} + \hat{k} \right) . \left( 5 \hat{i} - 9 \hat{j} + 2 \hat{k}\right) = 0\]
\[ \Rightarrow 5\lambda - 18 + 2 = 0\]
\[ \Rightarrow 5\lambda - 16 = 0\]
\[ \Rightarrow 5\lambda = 16\]
\[ \Rightarrow \lambda = \frac{16}{5}\]
APPEARS IN
RELATED QUESTIONS
Find the projection of the vector `hati+3hatj+7hatk` on the vector `2hati-3hatj+6hatk`
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`
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 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`
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
\[\vec{a} =\hat{i} - 2\hat{j} + \hat{k}\text{ and } \vec{b} = 4 \hat{i} - 4\hat{j} + 7 \hat{k}\]
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
\[\vec{a} = \lambda \hat{i} + 3 \hat{j} + 2 \hat{k}\text { and } \vec{b} = \hat{i} - \hat{j} + 3 \hat{k}\]
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 \[\vec{a} \text{ and } \vec{b} .\]
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 \[\vec{a} \text{ on } \vec{b}\]
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 \[\vec{a} \text{ and } \vec{b}\] are two vectors such that \[\left( \vec{a} + \vec{b} \right) . \left( \vec{a} - \vec{b} \right) = 0,\] find the relation between the magnitudes of \[\vec{a} \text{ and } \vec{b}\]
If \[\vec{a} = \hat{i} - \hat{j} \text{ and } \vec{b} = - \hat{j} + \hat{k} ,\] find the projection of \[\vec{a} \text{ on } \vec{b}\]
Write the value of \[\left( \vec{a} . \hat{i} \right) \hat{i} + \left( \vec{a} . \hat{j} \right) \hat{j} + \left( \vec{a} . \hat{k} \right) \hat{k} ,\] where \[\vec{a}\] is any vector.
Write the projection of \[\hat{i} + \hat{j} + \hat{k}\] along the vector \[\hat{j}\]
If \[\vec{a} \text{ and } \vec{b}\] are unit vectors, find the angle between \[\vec{a} + \vec{b} \text{ and } \vec{a} - \vec{b} .\]
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 \[\vec{a} = \hat{i} - \hat{j} + \hat{k} \text{ and } \vec{b} = \hat{i} + \hat{j} - \hat{k} .\]
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.
Write the angle between two vectors \[\vec{a} \text{ and } \vec{b}\] with magnitudes \[\sqrt{3}\] and 2 respectively if \[\vec{a} \cdot \vec{b} = \sqrt{6} .\]
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 \[2 \hat{i} + 6 \hat{j}+ 3 \hat{k} .\]
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 \[\vec{b}\]
Prove that, for any three vectors \[\vec{a} , \vec{b} , \vec{c}\] \[\left[ \vec{a} + \vec{b} , \vec{b} + \vec{c} , \vec{c} + \vec{a} \right] = 2 \left[ \vec{a} , \vec{b} , \vec{c} \right]\].
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 value of `hati(hatj + hatk)hatj * (hati + hatk) + hatk - (hati + hatj)` is-
If `θ` be the angle between any two vectors `veca` and `vecb`, then `|veca * vecb| = |veca xx vecb|`, when `θ` is equal to
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.
