Cross vs Dot Product
Explain the differences between a cross product and a dot product.
Expand Hint
The dot product, or scalar product, is an mathematical operation that takes two vectors and returns a single number. It is the projection of vector B onto vector A multiplied by the determinant of vector A. The dot product of vectors A and B:
$$$A\cdot B=a_xb_x+a_yb_y+a_zb_z=|A||B|cos\theta =B\cdot A$$$
where
$$|A|$$
is the determinant of vector A,
$$|B|$$
is the determinant of vector B, and:
$$A=a_x\textbf{i}+a_y\textbf{j}+a_z\textbf{k}$$
$$B=b_x\textbf{i}+b_y\textbf{j}+b_z\textbf{k}$$
Hint 2
The cross product, or vector product, is a mathematical operation that takes two vectors in 3D space and returns a new vector that is perpendicular (or normal) to the original two.
$$$A\times B= \left | \begin{matrix}\textbf{i} & \textbf{j} & \textbf{k}\\ a_x & a_y & a_z\\ b_x & b_y & b_z\end{matrix} \right |=|A||B|n\:sin\theta =-B\times A$$$
where
$$|A|$$
is the determinant of vector A,
$$|B|$$
is the determinant of vector B, n is the unit vector perpendicular to the plane of vectors A and B, and:
$$A=a_x\textbf{i}+a_y\textbf{j}+a_z\textbf{k}$$
$$$B=b_x\textbf{i}+b_y\textbf{j}+b_z\textbf{k}$$$
The dot product, or scalar product, is an mathematical operation that takes two vectors and returns a single number. It is the projection of vector B onto vector A multiplied by the determinant of vector A. The dot product of vectors A and B:
$$$A\cdot B=a_xb_x+a_yb_y+a_zb_z=|A||B|cos\theta =B\cdot A$$$
where
$$|A|$$
is the determinant of vector A,
$$|B|$$
is the determinant of vector B, and:
$$A=a_x\textbf{i}+a_y\textbf{j}+a_z\textbf{k}$$
$$B=b_x\textbf{i}+b_y\textbf{j}+b_z\textbf{k}$$
The cross product, or vector product, is a mathematical operation that takes two vectors in 3D space and returns a new vector that is perpendicular (or normal) to the original two.
$$$A\times B= \left | \begin{matrix}\textbf{i} & \textbf{j} & \textbf{k}\\ a_x & a_y & a_z\\ b_x & b_y & b_z\end{matrix} \right |=|A||B|n\:sin\theta =-B\times A$$$
where
$$|A|$$
is the determinant of vector A,
$$|B|$$
is the determinant of vector B, n is the unit vector perpendicular to the plane of vectors A and B, and:
$$A=a_x\textbf{i}+a_y\textbf{j}+a_z\textbf{k}$$
$$$B=b_x\textbf{i}+b_y\textbf{j}+b_z\textbf{k}$$$
In summary, the main differences:
- A cross product produces a vector while a dot product produces a number.
- The cross product only works in 3D while the dot product works in any number of dimensions.
- The cross product determines how much two vectors point in opposite directions, while the dot product determines how much two vectors point in identical directions.
A cross product produces a vector while a dot product produces a number. The cross product only works in 3D while the dot product works in any number of dimensions. The cross product determines how much two vectors point in opposite directions, while the dot product determines how much two vectors point in identical directions.
Time Analysis
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