Cross Product Calculator

An online calculator for finding the cross product of two vectors, with steps shown.

$$$\mathbf{\vec{u}}$$$: (
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$$$\mathbf{\vec{v}}$$$: (
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Hint: if you have two-dimensional vectors, set the third coordinates equal to or leave them empty.

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Your Input

Calculate $$$\left(3, 1, 4\right)\times \left(-2, 0, 5\right)$$$.

Solution

To find the cross product, we form a determinant the first row of which is a unit vector, the second row is our first vector, and the third row is our second vector: $$$\left|\begin{array}{ccc}\mathbf{\vec{i}} & \mathbf{\vec{j}} & \mathbf{\vec{k}}\\3 & 1 & 4\\-2 & 0 & 5\end{array}\right|.$$$

Now, just expand along the first row:

$$$\left|\begin{array}{ccc}\mathbf{\vec{i}} & \mathbf{\vec{j}} & \mathbf{\vec{k}}\\3 & 1 & 4\\-2 & 0 & 5\end{array}\right| = \left|\begin{array}{cc}1 & 4\\0 & 5\end{array}\right| \mathbf{\vec{i}} - \left|\begin{array}{cc}3 & 4\\-2 & 5\end{array}\right| \mathbf{\vec{j}} + \left|\begin{array}{cc}3 & 1\\-2 & 0\end{array}\right| \mathbf{\vec{k}} = \left(\left(1\right)\cdot \left(5\right) - \left(4\right)\cdot \left(0\right)\right) \mathbf{\vec{i}} - \left(\left(3\right)\cdot \left(5\right) - \left(4\right)\cdot \left(-2\right)\right) \mathbf{\vec{j}} + \left(\left(3\right)\cdot \left(0\right) - \left(1\right)\cdot \left(-2\right)\right) \mathbf{\vec{k}} = 5 \mathbf{\vec{i}} - 23 \mathbf{\vec{j}} + 2 \mathbf{\vec{k}}$$$

Thus, $$$\left(3, 1, 4\right)\times \left(-2, 0, 5\right) = \left(5, -23, 2\right)$$$.

Answer

$$$\left(3, 1, 4\right)\times \left(-2, 0, 5\right) = \left(5, -23, 2\right)$$$A