Write a vector of magnitude 12 units which makes 45º angle with x… - Vidyadip

Question

Write a vector of magnitude 12 units which makes 45º angle with x-axis, 60º angle with y-axis and an obtuse angle with z-axis.

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Answer

Suppose a vector $\vec{\text{r}}$ makes an angle 45º with OX, 60º with OY and having magnitude 12 units. $\text{l}=\cos45^{\circ}=\frac{1}{\sqrt2}$ and $\text{m}=\cos60^{\circ}=\frac{1}2$Now, $\text{l}^2+\text{m}^2+\text{n}^2=1$
$\Rightarrow\ \frac{1}2+\frac{1}4+\text{n}^2=1$
$\Rightarrow\ \text{n}^2=\frac{1}4$
$\Rightarrow\ =-\frac{1}2$ $[\because$ The angle with the z-axis is obtuse$]$
Therefore,
$\vec{\text{r}}=|\vec{\text{r}}|\big(\text{l}\hat{\text{i}}+\text{m}\hat{\text{j}}+\text{n}\hat{\text{k}}\big)$
$=12\Big(\frac{1}{\sqrt2}\hat{\text{i}}+\frac{1}2\hat{\text{j}}-\frac{1}2\hat{\text{k}}\Big)$
$=6\big(\sqrt2\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}\big)$

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