Advertisements
Advertisements
Question
Find the angle between the vectors with direction ratios proportional to 1, −2, 1 and 4, 3, 2.
Advertisements
Solution
\[\text{Let}\ \vec{a}\ \text{be a vector with direction ratios}\ 1, - 2, 1 . \]
\[ \Rightarrow \vec{a} =\hat{ i} - 2 \hat{j} + \hat {k} . \]
\[\text{Let} \ \vec{b}\ \text{be a vector with direction ratios} \ 4, 3, 2 . \]
\[ \Rightarrow \vec{b} = 4\hat{ i} + 3 \hat{j} + 2 \hat{k} . \]
\[\text{ Let }\ \theta \text{ be the angle between the given vectors } . \]
\[\text{ Now, } \]
\[\text{ cos }\theta = \frac{\vec{a} . \vec{b}}{\left| \vec{a} \right| \left| \vec{b} \right|} \]
\[ = \frac{\left( \hat{ i } - 2\hat { j } + \ \hat{k} \right) . \left( 4\hat { i } + 3 \ \hat{j}+ 2 \ \hat {k} \right)}{\left| \hat { i } - 2 \ \hat { j } +\ \hat {k} \right|\left| 4 \ \hat { i }+ 3\ \hat { j } + 2 \ \hat { k } \right|}\]
\[ = \frac{4 - 6 + 2}{\sqrt{1 + 4 + 1} \sqrt{16 + 9 + 4}} \]
\[ = \frac{0}{\sqrt{6} \sqrt{29}} \]
\[ = 0 \]
\[ \therefore \theta = \frac{\pi}{2}\]
\[\text{Thus, the angle between the given vectors measures }\frac{\pi}{2} .\]
APPEARS IN
RELATED QUESTIONS
Direction cosines of the line passing through the points A (- 4, 2, 3) and B (1, 3, -2) are.........
Write the direction ratios of the following line :
`x = −3, (y−4)/3 =( 2 −z)/1`
Find the angle between the lines whose direction ratios are 4, –3, 5 and 3, 4, 5.
If l1, m1, n1 and l2, m2, n2 are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m1n2 − m2n1, n1l2 − n2l1, l1m2 − l2m1.
If a line has direction ratios 2, −1, −2, determine its direction cosines.
Find the angle between the lines whose direction cosines are given by the equations
2l − m + 2n = 0 and mn + nl + lm = 0
What are the direction cosines of X-axis?
Write the distances of the point (7, −2, 3) from XY, YZ and XZ-planes.
Write the distance of the point (3, −5, 12) from X-axis?
Write the inclination of a line with Z-axis, if its direction ratios are proportional to 0, 1, −1.
Write the distance of the point P (x, y, z) from XOY plane.
Answer each of the following questions in one word or one sentence or as per exact requirement of the question:
Write the distance of a point P(a, b, c) from x-axis.
A parallelopiped is formed by planes drawn through the points (2, 3, 5) and (5, 9, 7), parallel to the coordinate planes. The length of a diagonal of the parallelopiped is
If P (3, 2, −4), Q (5, 4, −6) and R (9, 8, −10) are collinear, then R divides PQ in the ratio
If a line makes angles α, β, γ, δ with four diagonals of a cube, then cos2 α + cos2 β + cos2γ + cos2 δ is equal to
Find the direction cosines of the line joining the points P(4,3,-5) and Q(-2,1,-8) .
Verify whether the following ratios are direction cosines of some vector or not
`1/5, 3/5, 4/5`
Verify whether the following ratios are direction cosines of some vector or not
`4/3, 0, 3/4`
Find the direction cosines and direction ratios for the following vector
`3hat"i" - 3hat"k" + 4hat"j"`
Choose the correct alternative:
The unit vector parallel to the resultant of the vectors `hat"i" + hat"j" - hat"k"` and `hat"i" - 2hat"j" + hat"k"` is
Find the direction cosines of the line passing through the points P(2, 3, 5) and Q(–1, 2, 4).
The x-coordinate of a point on the line joining the points Q(2, 2, 1) and R(5, 1, –2) is 4. Find its z-coordinate.
P is a point on the line segment joining the points (3, 2, –1) and (6, 2, –2). If x co-ordinate of P is 5, then its y co-ordinate is ______.
If a line makes an angle of `pi/4` with each of y and z-axis, then the angle which it makes with x-axis is ______.
Find the equations of the two lines through the origin which intersect the line `(x - 3)/2 = (y - 3)/1 = z/1` at angles of `pi/3` each.
O is the origin and A is (a, b, c). Find the direction cosines of the line OA and the equation of plane through A at right angle to OA.
The area of the quadrilateral ABCD, where A(0,4,1), B(2, 3, –1), C(4, 5, 0) and D(2, 6, 2), is equal to ______.
The direction cosines of vector `(2hat"i" + 2hat"j" - hat"k")` are ______.
Find the direction cosine of a line which makes equal angle with coordinate axes.
The projections of a vector on the three coordinate axis are 6, –3, 2 respectively. The direction cosines of the vector are ______.
A directed line makes angles \[\alpha\], \[\beta\], and \[\gamma\] with the positive x-, y-, and z-axes respectively. What are these angles called?
Which identity must be satisfied by the direction cosines \[(l,m,n)\] of a line?
In the relation between direction ratios and direction cosines, what does the sign \[\pm\] in \[l=\pm\frac{a}{\sqrt{a^2+b^2+c^2}}\] depend on?
A line passes through \[P(x_1,y_1,z_1)\] and \[Q(x_2,y_2,z_2)\]. Which is one set of direction ratios of the line?
For points \[P(x_1,y_1,z_1)\] and \[Q(x_2,y_2,z_2)\], which expression gives \[PQ\]?
