Which of the following is the correct vector multiplication in terms of \(\hat{i}\), \(\hat{j}\) and \(\hat{k}\)?

î × ĵ = -k̂

î × ĵ = <span style="">k̂</span>

k̂ × î = -ĵ

k̂ × ĵ = î

Which of the following is the correct vector multiplication in terms of \(\hat{i}\), \(\hat{j}\) and \(\hat{k}\)? Correct Answer î × ĵ = <span style="">k̂</span>

Option (2)

CONCEPT:

Vector Product (Cross product): Its magnitude is equal to the products of the magnitude of two vectors and sine of the angle between them and whose direction is perpendicular to the plane of the two vectors.
- î, ĵ, and k̂ are the unit vectors whose magnitude of each of the vector is 1
- The angle betweenî, ĵ, and k̂ or any of the two of them is 90°.

EXPLANATION:

Vector product of orthogonal unit vectors, the magnitude of each of the vector î, ĵ, and k̂ is 1 or the angle between any of the two of them is 90°

î × ĵ = (1)(1) sin90° n̂ = n̂ (sin90° = 1)

As n̂ is a unit vector perpendicular to the plane of î and ĵ, so it just the third vector k̂

î × ĵ = k̂

Similarly, we can write ĵ × k̂ = î , k̂ × î = ĵ , ĵ × î = -k̂ , k̂ × ĵ = -i , î × k̂ = -ĵ

Additional Information

Aid to memory
- Write î, ĵ,k̂ cyclically around a cycle, as shown in the figure.

Multiplying two-unit vector anticlockwise, we get a positive value of the third unit vector

(î × ĵ = + k̂ ) and multiplying two-unit vector clockwise, we get the negative value of the third unit vector (ĵ × î = - î ),

we can also conclude this from the cross product is anti- commutative that is A × B = - B × A

ĵ × k̂ = î, k̂ × î = ĵ, ĵ × î = -k̂, k̂ × ĵ = -i , î × k̂ = -ĵ

মোঃ আরিফুল ইসলাম

Feb 20, 2025

Which of the following is the correct vector multiplication in terms of \(\hat{i}\), \(\hat{j}\) and \(\hat{k}\)?

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