Question

2.
The vectors Ā = 3î - 2j+k, B = -3; + 5k and C = 2ỉ + -4k form a
a) right angled triangle b) equilateral triangle c) isosceles triangle d) obtuse angled triangle ​

Answer 1

Answer

a ) Right angled triangle

Given

$\rm \vec{A}=3\hat{i}-2\hat{j}+k\\\\\rm \vec{B}=-3\hat{j}+5\hat{k}\\\\\rm \vec{C}=2\hat{i}-4\hat{k}$

To Find

Which type of triangle is formed by the given vectors

Key Points

While solving these type of questions , find magnitude of vectors and observe the magnitudes for finding type of triangle

Solution

Magnitude of A bar ,

$\rm |\vec{A}|=\sqrt{3^2+2^2+1^2}\\\\\to \rm |\vec{A}|=\sqrt{9+4+1}\\\\\to \rm |\vec{A}|=\sqrt{14}$

Likewise ,

$\rm |\vec{B}|=\sqrt{34}\ ,\ |\vec{C}|=\sqrt{20}$

By observing the magnitudes ,

$\rm |\vec{B}|^2=|\vec{A}|^2+|\vec{C}|^2\\\\\to \rm (\sqrt{34})^2=(\sqrt{14})^2+(\sqrt{20})^2\\\\\to \rm 34=14+20\\\\\to \rm 34=34$

The given vectors forms right angled triangle

So , Option A is correct

Answer 2

$\orange{\bigstar}\ \;$  Answer  $\green{\bigstar}$

$\rm \vec{A}=3\hat{i}-2\hat{j}+\hat{k}\\\\\rm \vec{B}=-3\hat{j}+5\hat{k}\\\\\rm \vec{C}=2\hat{i}-4\hat{k}$

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For a vector of $\rm\vec{P}= x\hat{i}+y\hat{j}+z\hat{k}$ ,

Magnitude is  $\rm |\vec{P}|=\sqrt{x^2+y^2+z^2}$

___________________

So ,

$\to \rm |\vec{A}|=\sqrt{14}\\\\\rm \to |\vec{B}|=\sqrt{34}\\\\\rm \to |\vec{C}|=\sqrt{20}$

Let's find squares of magnitudes of vectors ,

$\to \rm |\vec{A}|^2=14\\\\\to \rm |\vec{B}|^2=34\\\\\to \rm |\vec{C}|^2=20$

Since , $\rm |\vec{A}|^2+|\vec{C}|^2=|\vec{B}|^2$

So , the given vectors forms right angled triangle .

Option A is correct