Advertisements
Advertisements
Question
Find the acute angle between the planes `vec"r". (hat"i" - 2hat"j" - 2hat"k") = 1` and `vec"r". (3hat"i" - 6hat"j" - 2hat "k") = 0`
Advertisements
Solution
The vector equation of the planes is `vec"r". (hat"i" - 2hat"j" - 2hat"k") = 1` and `vec"r". (3hat"i" - 6hat"j" - 2hat"k") = 0`
It is known that if `vec"n"_1` and `vec"n"_2` are normal to the planes, `vec"r" . vec"n"_1 = "d"_1 "and" vec"r" . vec"n"_2 = "d"_2`, then the angle between them, is given by,
`costheta = |(vec("n"_1) . vec("n"_2))/(|vec("n"_1) |vec("n"_2)|)|`
So, the angle between the given planes will be
`costheta = |((hat"i"-2hat"j"-2hat"k").(3hat"i" -6hat"j" +2hat"k"))/((sqrt(1^2 + (-2)^2+(-2)^2))(sqrt(3^2+(-6)^2+(2)^2)))|`
`= |(3+12-4)/(3xx7)|`
`=|11/21|`
⇒ `theta = cos^-1|11/21|`
APPEARS IN
RELATED QUESTIONS
Can a vector have direction angles 45°, 60°, 120°?
A vector makes an angle of \[\frac{\pi}{4}\] with each of x-axis and y-axis. Find the angle made by it with the z-axis.
A vector \[\vec{r}\] is inclined at equal acute angles to x-axis, y-axis and z-axis.
If |\[\vec{r}\]| = 6 units, find \[\vec{r}\].
Find the direction cosines of the following vector:
`2hati + 2hatj - hatk`
Find the direction cosines of the following vectors:
\[6 \hat{i} - 2 \hat{j} - 3 \hat{k}\]
Find the direction cosines of the following vectors:
\[3 \hat{i} - 4 \hat{k}\]
Find the angles at which the following vectors are inclined to each of the coordinate axes:
\[4 \hat{i} + 8 \hat{j} + \hat{k}\]
Show that the vector \[\hat{i} + \hat{j} + \hat{k}\] is equally inclined with the axes OX, OY and OZ.
If a unit vector \[\vec{a}\] makes an angle \[\frac{\pi}{3}\] with \[\hat{i} , \frac{\pi}{4}\] with \[\hat{j}\] and an acute angle θ with \[\hat{k}\], then find θ and hence, the components of \[\vec{a}\].
A unit vector \[\overrightarrow{r}\] makes angles \[\frac{\pi}{3}\] and \[\frac{\pi}{2}\] with \[\hat{j}\text{ and }\hat{k}\] respectively and an acute angle θ with \[\hat{i}\]. Find θ.
What is the cosine of the angle which the vector \[\sqrt{2} \hat{i} + \hat{j} + \hat{k}\] makes with y-axis?
Write the direction cosines of the vector \[\hat{i} + 2 \hat{j} + 3 \hat{k}\].
Write the direction cosines of the vectors \[- 2 \hat{i} + \hat{j} - 5 \hat{k}\].
The vector cos α cos β \[\hat{i}\] + cos α sin β \[\hat{j}\] + sin α \[\hat{k}\] is a
If a line makes angles 90°, 45°, 135° with the X, Y and Z axes respectively, then its direction cosines are
The cosine of the angle included between the lines r = `(2hat"i" + hat"j" - 2hat"k") + lambda (hat"i" - 2hat"j" - 2hat"k")` and r = `(hat"i" + hat"j" + 3hat"k") + mu(3hat"i" + 2hat"j" - 6hat"k")` where λ, μ ∈ R is.
The direction cosines of a line which is perpendicular to lines whose direction ratios are 3, - 2, 4 and 1, 3, - 2 are ______.
A line lies in the XOZ plane and it makes an angle of 60° with the X-axis. Then its direction cosines are ______
The direction cosines of a line which is perpendicular to both the lines whose direction ratios are -2, 1, -1 and -3, -4, 1 are ______
The angle between the lines whose direction cosines satisfy the equations l + m + n = 0 and l2 = m2 + n2 is ______.
