\(\vec{r} = (\cos\theta)\hat{i} + (\sin\theta)\hat{j} + \hat{k}\), how much angle does this make with the positive direction of \(z\)-axis?
\(\pi/4\)
The angle \(\phi\) that \(\vec{r}\) makes with the \(z\)-axis satisfies \(\cos\phi=\dfrac{\vec{r}\cdot\hat{k}}{|\vec{r}|}=\dfrac{1}{\sqrt{\cos^2\theta+\sin^2\theta+1}}=\dfrac{1}{\sqrt{2}}\), so \(\phi=\pi/4\).
Similar Questions
A point on a line has coordinates (p + 1, p - 3, √2p) where p is any real number. What are the direction cosines of the line?
If L is the line with direction ratios < 3, -2, 6 > and passing through (1, -1, 1), then what are the coordinates of the points on L whose distance from (1, -1, 1) is 2 units?
If l, m, n are the direction cosines of the line x - 1 = 2(y + 3) = 1 - z, then what is l 4+ m 4+ n 4equal to?
A plane cuts intercepts 2, 2, 1 on the coordinate axes. What are the direction cosines of the normal to the plane?
Consider the following statements :
1. The direction ratios of y-axis can be <0, 4, 0>
2. The direction ratios of a line perpendicular to z-axis can be <5, 6, 0>
Which of the statements given above is/are correct?
The direction ratios of the line perpendicular to the lines with direction ratios < 1, -2, -2 > and < 0, 2, 1 > are
A straight line with direction cosines \(\left\langle {0,\;1,\;0} \right\rangle\) is
If a line has direction ratios < a + b, b + c, c + a >, then what is the sum of the squares of its direction cosines?
What is cos2β + cos2γ equal to ?
What are the direction cosines of z-axis?
Important Questions from Direction ratios and Direction cosines
A point on a line has coordinates (p + 1, p - 3, √2p) where p is any real number. What are the direction cosines of the line?
If L is the line with direction ratios < 3, -2, 6 > and passing through (1, -1, 1), then what are the coordinates of the points on L whose distance from (1, -1, 1) is 2 units?
If l, m, n are the direction cosines of the line x - 1 = 2(y + 3) = 1 - z, then what is l 4+ m 4+ n 4equal to?
A plane cuts intercepts 2, 2, 1 on the coordinate axes. What are the direction cosines of the normal to the plane?
Which of the following is the direction cosines of z, y and x-axis?
