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Write two different vectors having same direction.

1 Answer

  • Two vectors are said to be equal if they have the same direction,but their magnitude can be different.
  • Direction cosines of a vector are $\bigg(\large\frac{a_1}{\sqrt{a_1^2+a_2^2+a_3^2}},\large\frac{a_2}{\sqrt{a_1^2+a_2^2+a_3^2}},\large\frac{a_3}{\sqrt{a_1^2+a_2^2+a_3^2}}\bigg)$
Step 1:
Let us consider two vectors $\overrightarrow{a}=\hat i+\hat j+\hat {k}$ and $\overrightarrow b=3\hat i+3\hat j+3\hat k$
$\mid\overrightarrow a\mid=\sqrt{1^2+1^2+1^2}$
$\qquad=\sqrt 3$
$\mid\overrightarrow b\mid=\sqrt{3^2+3^2+3^2}$
$\qquad=\sqrt 27$
$\qquad=3\sqrt 3$
Hence $\overrightarrow{a}$ and $\overrightarrow{b}$ are different in magnitude.
Step 2:
The direction cosines of $\overrightarrow{a}$ are $\big(\large\frac{1}{\sqrt 3},\large\frac{1}{\sqrt 3},\large\frac{1}{\sqrt 3}\big)$
The direction cosines of $\overrightarrow{b}$ are $\big(\large\frac{3}{\sqrt {27}},\large\frac{3}{\sqrt {27}},\large\frac{3}{\sqrt {27}}\big)$
(i.e)The direction cosines of $\overrightarrow{b}$ are $\big(\large\frac{1}{\sqrt {3}},\large\frac{1}{\sqrt {3}},\large\frac{1}{\sqrt {3}}\big)$
Clearly $\overrightarrow{a}$ and $\overrightarrow{b}$ are same in direction.
Step 3:
Hence vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ have the same direction but different magnitude.
answered May 17, 2013 by sreemathi.v

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