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Vector Algebra question

2003 · Shift 0 · Q90

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JEE Main
Mathematics
Vector Algebra
2003 · Shift 0 · Q90

Vector Algebra question

2003 · Shift 0 · Q90

JEE MainMathematicsVector AlgebraMCQ+4 / −1

Let

\overrightarrow u = \widehat i + \widehat j,\,\overrightarrow v = \widehat i - \widehat j

and

\overrightarrow w = \widehat i + 2\widehat j + 3\widehat k\,\,.

If

\widehat n

is a unit vector such that

\overrightarrow u .\widehat n = 0

and

\overrightarrow v .\widehat n = 0\,\,,

then

\left| {\overrightarrow w .\widehat n} \right|

is equal to :

A
$3$
B
$0$
C
$1$
D
$2$

View written solutionFree

Correct answer: A

Write the given vectors in component form:

\vec u = (1,1,0), \quad \vec v = (1,-1,0), \quad \vec w = (1,2,3)

We need a unit vector $\hat n=(x,y,z)$ such that

\vec u \cdot \hat n = 0, \qquad \vec v \cdot \hat n = 0.

Apply the orthogonality conditions:

From $\vec u \cdot \hat n=0$ ,

x+y=0

From $\vec v \cdot \hat n=0$ ,

x-y=0

Solve these equations:

Adding both equations,

2x=0 \implies x=0

Then from $x+y=0$ ,

y=0

So,

\hat n=(0,0,z)

Since $\hat n$ is a unit vector,

|\hat n|=1 \implies z=\pm 1

Hence,

\hat n=\pm \hat k

Now compute $\vec w\cdot \hat n$ :

\vec w\cdot \hat n = (1,2,3)\cdot(0,0,\pm 1)=\pm 3

Therefore,

\left|\vec w\cdot \hat n\right|=3

Check options:

A: $3$ ✅
B: $0$ ❌
C: $1$ ❌
D: $2$ ❌

So the correct answer is A.

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