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Vector (or Cross) Product of Two Vectors is considered one the most difficult concept.
Vector Product in Terms of Components is considered one of the most asked concept.
55 Questions around this concept.
Let and . If is a unit vector such that and then is equal to:
If and the angle between is equal to
are two vectors and is a vector such that then
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Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{b}|=1$ and $|\vec{b} \times \vec{a}|=2$. Then $|(\vec{b} \times \vec{a})-\vec{b}|^2$ is equal to
Let $\overrightarrow{\mathrm{a}}=-5 \hat{\mathrm{i}}+\hat{\mathrm{j}}-3 \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-4 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}=(((\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) \times \hat{\mathrm{i}}) \times \hat{\mathrm{i}}) \times \hat{\mathrm{i}}$. Then $\overrightarrow{\mathrm{c}} \cdot(-\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})$ is equal to :
The vector product of two nonzero vectors and , is denoted by and defined as,
where Ө is the angle between and , 0 ≤ θ ≤ π and is a unit vector perpendicular to both and , such that , and form a right hand system.
Observe that, the direction of is opposite to that of as shown in the figure.
i.e.
So the vector product is not commutative.
Properties of Vector Product
Proof:
Angle between Two Vectors
If Ө is the angle between the vectors and , then
Vector perpendicular to the plane of two given vectors
The unit vector perpendicular to the plane of and is
Also note that is also a unit vector perpendicular to the plane of and .
Vectors of magnitude ‘λ’ perpendicular to the plane of and are given by
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