Cross Product of Vectors Quiz

The angle between two perpendicular vectors is 90°.

The dot product of two perpendicular vectors is equal to 0.

The cross product of the unit vectors $\hat{i}$, $\hat{j}$, and $\hat{k}$ follows similar rules as the cross product of vectors.

The angle between the same vectors is equal to 0°, and hence their cross product is equal to 0.

The cross product of two perpendicular vectors gives a vector, which is perpendicular to the two original vectors.

The cross product $\vec{u} \times \vec{v}$ is a vector perpendicular to the plane containing $\vec{u}$ and $\vec{v}$, with magnitude equal to $|\vec{u}||\vec{v}|\sin \theta$, where $\theta$ is the angle between $\vec{u}$ and $\vec{v}$. The direction of the cross product is given by the right-hand rule.

The cross product of two parallel vectors is zero.

The cross product results in a vector perpendicular to the plane containing the two vectors, while the dot product results in a scalar quantity. Additionally, the dot product is commutative ($\vec{u} \cdot \vec{v} = \vec{v} \cdot \vec{u}$), but the cross product is not ($\vec{u} \times \vec{v} = -\vec{v} \times \vec{u}$).

The magnitude of the cross product of two vectors $\vec{u}$ and $\vec{v}$ is equal to the area of the parallelogram formed by $\vec{u}$ and $\vec{v}$, i.e., $|\vec{u} \times \vec{v}| = |\vec{u}||\vec{v}|\sin \theta$, where $\theta$ is the angle between $\vec{u}$ and $\vec{v}$.

The cross product of two non-parallel vectors lying in a plane gives a vector perpendicular to that plane. Therefore, if $\vec{u}$ and $\vec{v}$ are two non-parallel vectors in a plane, then $\vec{u} \times \vec{v}$ is the normal vector to that plane.

The right-hand rule states that if you point the fingers of your right hand in the direction of the first vector $\vec{u}$ and curl them towards the second vector $\vec{v}$, then your thumb points in the direction of the cross product $\vec{u} \times \vec{v}$.

The two products of vectors are the dot product (scalar product) and the cross product (vector product).

Geometrically, the dot product of two vectors $\mathbf{v}$ and $\mathbf{w}$ is equal to the product of the magnitudes of the two vectors and the cosine of the angle between them: $\mathbf{v} \cdot \mathbf{w} = |\mathbf{v}| |\mathbf{w}| \cos \theta$. Algebraically, the dot product is defined as $\mathbf{v} \cdot \mathbf{w} = v_1 w_1 + v_2 w_2 + v_3 w_3$.

The dot product of two vectors is a scalar quantity, while the cross product of two vectors is a vector quantity. The dot product is used to calculate the magnitude of the projection of one vector onto another, while the cross product is used to find a vector that is perpendicular to both of the original vectors.

The zero vector is a vector with zero magnitude and no direction. It is denoted by $\mathbf{0}$ (0,0,0). The zero vector is the additive identity of vectors, meaning that when added to any other vector, the result is the original vector.

The dot product of two vectors $\mathbf{v}$ and $\mathbf{w}$ is equal to the product of the magnitudes of the two vectors and the cosine of the angle between them: $\mathbf{v} \cdot \mathbf{w} = |\mathbf{v}| |\mathbf{w}| \cos \theta$. This means that the dot product is maximized when the two vectors are parallel (cos $\theta$ = 1) and minimized when the two vectors are perpendicular (cos $\theta$ = 0).

The cross product of two vectors $\mathbf{v}$ and $\mathbf{w}$ is a vector that is perpendicular to both $\mathbf{v}$ and $\mathbf{w}$. The direction of the cross product is determined by the right-hand rule, and the magnitude of the cross product is given by $|\mathbf{v} \times \mathbf{w}| = |\mathbf{v}| |\mathbf{w}| \sin \theta$, where $\theta$ is the angle between $\mathbf{v}$ and $\mathbf{w}$.