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Direction Cosines and Direction Ratio is considered one of the most asked concept.
19 Questions around this concept.
A vector $\vec{\gamma}$ is such that $\vec{\gamma}=3 \hat{i}-6 \hat{j}+2 \hat{k}$ then $\cos \alpha+\cos \beta+\cos \gamma$ equals:
For P(1,2,3); the direction cosines of OP where O is the origin is:
The direction cosines of the vector $3i-4j+5k$ are
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If the coordinates of the points A,B,C be (-1,3,2),(2,3,5) and (3,5,-2) respectively, then $\angle A=$
Two lines whose direction rations are $<a_1, b_1, c_1>$ and $<a_2, b_2, c_2>$ respectively are perpendicular if
A line making $30^{\circ}, 60^{\circ}, 90^{\circ}$ with positive direction of $x, y, z$ axes respectively has direction angles as:
Let $\mathrm{P}(3,2,3), \mathrm{Q}(4,6,2)$ and $\mathrm{R}(7,3,2)$ be the vertices of $\triangle \mathrm{PQR}$. Then, the angle $\angle \mathrm{QPR}$ is
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Direction Cosines (DCs)
Let $r$ be the position vector of a point $P(x, y, z)$. Then, the direction cosines of vector $r$ are the cosines of angles $\alpha, \beta$, and $y$ (i.e. $\cos \alpha$, $\cos \beta$, and $\cos \gamma$ ) that the vector $r$ makes with the positive direction of $X, Y$, and $Z$-axes respectively. Direction cosines are usually denoted by $I, m$, and $n$ respectively.
From the figure, note that ΔOAP is a right-angled triangle and thus, we have
$
\cos \alpha=\frac{x}{r}(r \text { stands for }|r|)
$
Similarly, from the right angled triangles $O B P$ and $O C P$, We have,
$
\cos \beta=\frac{y}{r} \text { and } \cos \gamma=\frac{z}{r}
$
So we have the following results,
$
\begin{aligned}
& \cos \alpha=l=\frac{x}{\sqrt{x^2+y^2+z^2}}=\frac{x}{|\mathbf{r}|}=\frac{x}{r} \\
& \cos \beta=m=\frac{y}{\sqrt{x^2+y^2+z^2}}=\frac{y}{|\mathbf{r}|}=\frac{y}{r} \\
& \cos \gamma=n=\frac{z}{\sqrt{x^2+y^2+z^2}}=\frac{z}{|\mathbf{r}|}=\frac{z}{r}
\end{aligned}
$
Also,
$
\begin{aligned}
& l^2=\frac{x^2}{x^2+y^2+z^2} \\
& m^2=\frac{y^2}{x^2+y^2+z^2} \\
& n^2=\frac{z^2}{x^2+y^2+z^2}
\end{aligned}
$
Add (i), (ii) and (iii)
$
\begin{aligned}
& l^2+m^2+n^2=\frac{x^2}{x^2+y^2+z^2}+\frac{y^2}{x^2+y^2+z^2}+\frac{z^2}{x^2+y^2+z^2} \\
& \Rightarrow l^2+m^2+n^2=1 \\
& \Rightarrow \cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1
\end{aligned}
$
The coordinates of the point P may also be expressed as (lr, mr, nr).
Direction ratios (DRs)
Direction Ratios are any set of three numbers that are proportional to the Direction cosines.
If $\mathrm{I}, \mathrm{m}, \mathrm{n}$ are DCs of a vector then $\lambda l, \lambda m, \lambda n$ are DRs of this vector, where a can take any real value.
DRs are also denoted as a, b, and c, respectively.
A vector has only one set of DCs, but infinite sets of DRs.
Note:
The coordinates of a point equal lr, Mr, and nr, which are proportional to the direction cosines. Hence the coordinates of a point are also its DRs.
If $\vec{r}=ai+b \hat{j}+c \hat{k}$, then $a, b$ and $c$ are one of the direction ratios of the given vector. Also, if $a^2+b^2+c^2=1$, then $a, b$ and $c$ will be direction cosines of the given vector.
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