Advertisements
Advertisements
Question
Advertisements
Solution
Given: \[ \vec{a} = 3 \hat{i} - \hat{j} - 4 \hat{k}, \vec{b} = - 2 \hat{i} + 4 \hat{j} - 3 \hat{k}\text{ and } \vec{c} = \hat{i} + 2 \hat{j} - \hat{k}\].
\[\text { Now } , 3 \vec{a} - 2 \vec{b} + 4 \vec{c} = 3\left( 3 \hat{i} - \hat{j} - 4 \hat{k} \right) - 2\left( - 2 \hat{i} + 4 \hat{j} - 3 \hat{k} \right) + 4\left( \hat{i} + 2 \hat{j} - \hat{k} \right)\]
\[ = 9 \hat{i} - 3 \hat{j} - 12\hat{k} + 4 \hat{i} - 8 \hat{j} + 6 \hat{k} + 4 \hat{i} + 8 \hat{j} - 4 \hat{k} \]
\[ = 17 \hat{i} - 3 \hat{j} - 10 \hat{k}\]
Hence,
\[\left| 3 \vec{a} - 2 \vec{b} + 4 \vec{c} \right| = \sqrt{{17}^2 + \left( - 3 \right)^2 + \left( - 10 \right)^2} = \sqrt{289 + 9 + 100} = \sqrt{398}\]
APPEARS IN
RELATED QUESTIONS
Write the position vector of the point which divides the join of points with position vectors `3veca-2vecb and 2veca+3vecb` in the ratio 2 : 1.
Find the position vector of a point which divides the join of points with position vectors `veca-2vecb" and "2veca+vecb`externally in the ratio 2 : 1
If `bara, barb, barc` are position vectors of the points A, B, C respectively such that `3bara+ 5barb-8barc = 0`, find the ratio in which A divides BC.
Classify the following measures as scalar and vector.
10 kg
Classify the following as scalar and vector quantity.
Time period
`veca and -veca` are collinear.
Two collinear vectors are always equal in magnitude.
Two collinear vectors having the same magnitude are equal.
Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are `hati + 2hatj - hatk` and `-hati + hatj + hatk` respectively, externally in the ratio 2:1.
Express \[\vec{AB}\] in terms of unit vectors \[\hat{i}\] and \[\hat{j}\], when the points are A (4, −1), B (1, 3)
Find \[\left| \vec{A} B \right|\] in each case.
Express \[\vec{AB}\] in terms of unit vectors \[\hat{i}\] and \[\hat{j}\], when the points are A (−6, 3), B (−2, −5)
Find \[\left| \vec{A} B \right|\] in each case.
ABCD is a parallelogram. If the coordinates of A, B, C are (−2, −1), (3, 0) and (1, −2) respectively, find the coordinates of D.
Dot product of a vector with \[\hat{i} + \hat{j} - 3\hat{k} , \hat{i} + 3\hat{j} - 2 \hat{k} \text{ and } 2 \hat{i} + \hat{j} + 4 \hat{k}\] are 0, 5 and 8 respectively. Find the vector.
The adjacent sides of a parallelogram are represented by the vectors \[\vec{a} = \hat{i} + \hat{j} - \hat{k}\text{ and }\vec{b} = - 2 \hat{i} + \hat{j} + 2 \hat{k} .\]
Find unit vectors parallel to the diagonals of the parallelogram.
If \[\hat{ a } \text{ and } \hat{b }\] are unit vectors inclined at an angle θ, prove that
\[\tan\frac{\theta}{2} = \frac{\left| \hat{a} -\hat{b} \right|}{\left| \hat{a} + \hat{b} \right|}\]
Show that the vectors \[\vec{a} = \frac{1}{7}\left( 2 \hat{i} + 3 \hat{j} + 6 \hat{k} \right), \vec{b} = \frac{1}{7}\left( 3\hat{i} - 6 {j} + 2 \hat{k} \right), \vec{c} = \frac{1}{7}\left( 6 \hat{i} + 2 \hat{j} - 3 {k} \right)\] are mutually perpendicular unit vectors.
If \[\vec{p} = 5 \hat{i} + \lambda \hat{j} - 3 \hat{k} \text{ and } \vec{q} = \hat{i} + 3 \hat{j} - 5 \hat{k} ,\] then find the value of λ, so that \[\vec{p} + \vec{q}\] and \[\vec{p} - \vec{q}\] are perpendicular vectors.
If either \[\vec{a} = \vec{0} \text{ or } \vec{b} = \vec{0}\] then \[\vec{a} \cdot \vec{b} = 0 .\] But the converse need not be true. Justify your answer with an example.
Show that the vectors \[\vec{a} = 3 \hat{i} - 2 \hat{j} + \hat{k} , \vec{b} = \hat{i} - 3 \hat{j} + 5 \hat{k} , \vec{c} = 2 \hat{i} + \hat{j} - 4 \hat{k}\] form a right-angled triangle.
If \[\vec{a} = 2 \hat{i} + 2 \hat{j} + 3 \hat{k} , \vec{b} = - \hat{i} + 2 \hat{j} + \hat{k} \text{ and } \vec{c} = 3 \hat{i} + \hat{j}\] \[\vec{a} + \lambda \vec{b}\] is perpendicular to \[\vec{c}\] then find the value of λ.
If the vertices A, B and C of ∆ABC have position vectors (1, 2, 3), (−1, 0, 0) and (0, 1, 2), respectively, what is the magnitude of ∠ABC?
Find the value of x for which \[x \left( \hat{i} + \hat{j} + \hat{k} \right)\] is a unit vector.
If \[\vec{a} = \hat{i} + \hat{j} + \hat{k} , \vec{b} = 2 \hat{i} - \hat{j} + 3 \hat{k} \text{ and }\vec{c} = \hat{i} - 2 \hat{j} + \hat{k} ,\] find a unit vector parallel to \[2 \vec{a} - \vec{b} + 3 \vec{c .}\]
Show that the vectors \[2 \hat{i} - 3 \hat{j} + 4 \hat{k}\text{ and }- 4 \hat{i} + 6 \hat{j} - 8 \hat{k}\] are collinear.
If \[\vec{a} \times \vec{b} = \vec{c} \times \vec{d} \text { and } \vec{a} \times \vec{c} = \vec{b} \times \vec{d}\] , show that \[\vec{a} - \vec{d}\] is parallel to \[\vec{b} - \vec{c}\] where \[\vec{a} \neq \vec{d} \text { and } \vec{b} \neq \vec{c}\] .
if `hat"i" + hat"j" + hat"k", 2hat"i" + 5hat"j", 3hat"i" + 2 hat"j" - 3hat"k" and hat"i" - 6hat"j" - hat"k"` respectively are the position vectors A, B, C and D, then find the angle between the straight lines AB and CD. Find whether `vec"AB" and vec"CD"` are collinear or not.
The unit normal to the plane 2x + y + 2z = 6 can be expressed in the vector form as
If `veca, vecb, vecc` are vectors such that `[veca, vecb, vecc]` = 4, then `[veca xx vecb, vecb xx vecc, vecc xx veca]` =
Assertion (A): If a line makes angles α, β, γ with positive direction of the coordinate axes, then sin2 α + sin2 β + sin2 γ = 2.
Reason (R): The sum of squares of the direction cosines of a line is 1.
A line l passes through point (– 1, 3, – 2) and is perpendicular to both the lines `x/1 = y/2 = z/3` and `(x + 2)/-3 = (y - 1)/2 = (z + 1)/5`. Find the vector equation of the line l. Hence, obtain its distance from the origin.
Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are `hati + 2hatj - hatk` and `-hati + hatj + hatk` respectively, internally the ratio 2:1.
