Advertisements
Advertisements
प्रश्न
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.
उत्तर
Let the vector \[\overrightarrow{OP}\] makes an angle \[\alpha = 45^{\circ} \text{ and }\beta = 45^{\circ}\] with OX, OY respectively. Suppose \[\overrightarrow{OP}\] is inclined at angle \[\gamma\] to OZ
Let l, m, n be the direction cosines of \[\overrightarrow{OP}\].
Then,
\[l = \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}}\]
\[m = \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}}\]
\[n = \cos \gamma\]
Now, we have,
\[l^2 + m^2 + n^2 = 1\]
\[ \Rightarrow \frac{1}{2} + \frac{1}{2} + n^2 = 1\]
\[ \Rightarrow n^2 = 0\]
\[ \Rightarrow n = 0\]
\[ \Rightarrow \cos \gamma = \cos \frac{\pi}{2}\]
\[ \Rightarrow \gamma = \frac{\pi}{2}\]
Hence, the angle made by it with the \[z -\] axis is \[\frac{\pi}{2}\]
APPEARS IN
संबंधित प्रश्न
Can a vector have direction angles 45°, 60°, 120°?
Prove that 1, 1, 1 cannot be direction cosines of a straight line.
A vector \[\vec{r}\] is inclined at equal acute angles to x-axis, y-axis and z-axis.
If |\[\vec{r}\]| = 6 units, find \[\vec{r}\].
A vector \[\vec{r}\] is inclined to -axis at 45° and y-axis at 60°. If \[|\vec{r}|\] = 8 units, find \[\vec{r}\].
Find the direction cosines of the following vectors:
\[2 \hat{i} + 2 \hat{j} - \hat{k}\]
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:
\[\hat{j} - \hat{k}\]
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.
Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are \[\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} .\]
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 \[\vec{a}\].
Write the direction cosines of the vector \[\overrightarrow{r} = 6 \hat{i} - 2 \hat{j} + 3 \hat{k} .\]
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 two different vectors having same direction.
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 direction co-sines of the line which bisects the angle between positive direction of Y and Z axes 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 ______.
Which of the following can not be the direction cosines of a line?
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 ______.
