Question

The vector having a magnitude of 10 and perpendicular to the vector 3î - 4j is (A) 4î +3 (B) 5√2î-5√2j(C) 8î+6j (D) 8î -6j​

Answer 1

Answer:

c

Explanation:

the angle between any two vectors is given by A.B =|A||B|cos(theta)

given both vectors are perpendicular so angle between them 90° so theta =90°

A.B = 0

by option verification

(3î - 4j ).( 8î+6j )= 24-24 = 0

so 8î+6j is the vector which is perpendicular to given vector

Answer 2

Given:-

• Magnitude of vector = 10
• Vector = 3î - 4ĵ

To Find:-

• The vector perpendicular to 3î - 4ĵ having a magnitude of 10

Solution:-

We know,

• When two vectors are perpendicular to each other, their dot product is always 0. ( statement a)

Let:-

• The other vector be B = (pî + qĵ)

Now,

(3î - 4ĵ) . (pî + qĵ) = 0

= 3p - 4q = 0

=> 3p = 4q

=> p = 4q/3 .... (i)

It is given that:

• |B| = 10

This means:-

√p² + q² = 10

Squaring both the sides:

= p² + q² = 100

Putting the value of p from equation (i)

= (4q/3)² + q² = 100

= 16q²/9 + q² = 100

= (16q² + 9q²)/9 = 100

= 25q² = 9 × 100

= q² = 900/25

= q² = 36

= q = √36 = ±6

Now,

Putting the value of q in equation (i)

= p = 4q/3

= p = 4(±6)/3

= p = ±8

Therefore,

The possible vectors for (pî + qĵ) are as follows:-

• (-8î - 6ĵ)
• (8î + 6ĵ)
• (-8î + 6ĵ)
• (8î - 6ĵ)

Now let us find which vector is exactly perpendicular to the vector (3î - 4ĵ)

• According to statement a, the dot product of the two vectors should be zero for the two vectors to be perpendicular.

Hence,

• For first vector {i.e. (-8î - 6ĵ)}

Dot product with (3î - 4ĵ)

= (3î - 4ĵ) . (-8î - 6ĵ)

= 3 × (-8) + {(-4) × (-6)}

= -24 + 24

= 0

∴ Vector (-8î - 6ĵ) is perpendicular to vector (3î - 4ĵ)

• For second vector {i.e. (8î + 6ĵ)}

Dot product with (3î - 4ĵ)

= (3î - 4ĵ) . (8î + 6ĵ)

= 3 × 8 + {(-4) × 6}

= 24 - 24

= 0

∴ Vector (8î + 6ĵ) is also perpendicular to vector (3î - 4ĵ)

• For third vector {i.e. (-8î + 6ĵ)}

Dot product with (3î - 4ĵ)

= (3î - 4ĵ) . (-8î + 6ĵ)

= 3 × (-8) + (-4) × 6

= -24 - 24

= -48

∴ Vector (-8î + 6ĵ) is not perpendicular to vector (3î - 4ĵ)

• For forth vector {i.e (8î - 6ĵ)}

Dot product with (3î - 4ĵ)

= (3î - 4ĵ) . (8î - 6ĵ)

= (3 × 8) + {(-4) × (-6)}

= 24 + 24

= 48

∴ Vector (8î - 6ĵ) is also not perpendicular to vector (3î - 4ĵ).

• Hence, we can clearly see that there are only two possible vectors which can be perpendicular to vector (3î - 4ĵ)

• From the above calculations the vector in option (d) gets eliminated as it is a vector which is not perpendicular to vector (3î - 4ĵ)

Therefore the answer is option (c) (8î + 6ĵ)

________________________________