If vectors v ⃗ = ⟨ 3 , 1 , 0 ⟩ v ⃗=⟨3,1,0⟩ v ⃗ = ⟨ 3 , 1 , 0 ⟩ , u ⃗ = ⟨ 0 , − 2 , 0 ⟩ u ⃗=⟨0,-2,0⟩ u ⃗ = ⟨ 0 , − 2 , 0 ⟩ , and w ⃗ = v ⃗ × u ⃗ w ⃗=v ⃗×u ⃗ w ⃗ = v ⃗ × u ⃗ , find w ⃗⃗ w⃗⃗ w ⃗⃗ .

w ⃗ = ⟨ 0 , 0 , − 6 ⟩ w⃗=\langle0,0,-6\rangle w ⃗ = ⟨ 0 , 0 , − 6 ⟩

w ⃗ = ⟨ 0 , − 2 , 0 ⟩ w⃗=\langle0,-2,0\rangle w ⃗ = ⟨ 0 , − 2 , 0 ⟩

w ⃗ = ⟨ 0 , 0 , 6 ⟩ w⃗=\langle0,0,6\rangle w ⃗ = ⟨ 0 , 0 , 6 ⟩

w ⃗ = ⟨ 0 , 0 , − 2 ⟩ w⃗=\langle0,0,-2\rangle w ⃗ = ⟨ 0 , 0 , − 2 ⟩

If vectors a ⃗ = 5 ı ^ a⃗=5î a ⃗ = 5 ı ^ , b ⃗ = 12 k ^ b⃗=12k̂ b ⃗ = 12 k ^ and c ⃗ = a ⃗ × b ⃗ c⃗=a⃗\times b⃗ c ⃗ = a ⃗ × b ⃗ , find c ⃗ c⃗ c ⃗ .

c ⃗ = − 60 ȷ ^ c⃗=-60ĵ c ⃗ = − 60 ȷ ^

c ⃗ = 60 ȷ ^ c⃗=60ĵ c ⃗ = 60 ȷ ^

c ⃗ = 5 ı ^ + 12 k ^ c⃗=5î+12k̂ c ⃗ = 5 ı ^ + 12 k ^

c ⃗ = 60 k ^ c⃗=60k̂ c ⃗ = 60 k ^

What are the properties of the cross product?

The cross product has several important properties: 1) It is anti-commutative, meaning a × b = - b × a . 2) It is distributive over vector addition, meaning a × ( b + c ) = a × b + a × c . 3) The cross product of parallel vectors is zero. 4) The magnitude of the cross product is equal to the area of the parallelogram formed by the two vectors.

14. Vectors

Cross Product

14. Vectors

Cross Product: Videos & Practice Problems

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Topic summary

The cross product of two vectors results in a new vector that is perpendicular to the original vectors. To compute the cross product, set up a matrix with the unit vectors i, j, k in the first row, followed by the components of the vectors. Use the determinant method to find each component by multiplying and subtracting the appropriate elements. The resulting vector can be expressed in component form. Recognizing the pattern of multiplying unlike components simplifies the process, making it easier to derive the final vector result.

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Computing the Cross Product

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Computing the Cross Product Video Summary

The cross product is a mathematical operation that takes two vectors and produces a third vector that is perpendicular to the plane formed by the original vectors. Unlike the dot product, which results in a scalar, the cross product yields a vector. This operation is particularly useful in physics and engineering, where understanding the orientation of vectors is crucial.

To compute the cross product of two vectors, say $\mathbf{u}$ and $\mathbf{v}$, we can follow a systematic approach using a determinant of a matrix. For vectors $\mathbf{u} = (u_1, u_2, u_3)$ and $\mathbf{v} = (v_1, v_2, v_3)$, we can set up the following matrix:

\[$\begin{vmatrix}\[\mathbf{i}$ & $\mathbf{j}$ & $\mathbf{k}$ \$\u$_1 & u_2 & u_3 \$\v$_1 & v_2 & v_3$\end{vmatrix}\]\end{equation}$

Here, $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ are the unit vectors in the x, y, and z directions, respectively. The first row contains the unit vectors, while the second and third rows contain the components of vectors $\mathbf{u}$ and $\mathbf{v}$.

The next step involves calculating the determinant using a cross-multiplication pattern. For each component of the resulting vector $\mathbf{w} = \mathbf{u} \times \mathbf{v}$, we can derive the following:

1. The x-component ($w_1$) is calculated as:

\[w_1 = u_2 v_3 - u_3 v_2$\end{equation}$

2. The y-component ($w_2$) is given by:

\[w_2 = u_3 v_1 - u_1 v_3$\end{equation}$

3. The z-component ($w_3$) is determined by:

\[w_3 = u_1 v_2 - u_2 v_1$\end{equation}$

By applying these formulas, we can find the components of the resulting vector. For example, if we have $\mathbf{u} = (2, 0, 1)$ and $\mathbf{v} = (0, -1, 2)$, we can compute:

\[w_1 = 0 $\cdot$ 2 - 1 $\cdot$ (-1) = 1$\end{equation}$

\[w_2 = 1 $\cdot$ 0 - 2 $\cdot$ 2 = -4$\end{equation}$

\[w_3 = 2 $\cdot$ (-1) - 0 $\cdot$ 0 = -2$\end{equation}$

Thus, the cross product $\mathbf{w}$ can be expressed in vector form as:

\[$\mathbf{w}$ = (1, -4, -2)$\end{equation}$

In summary, the cross product is a powerful tool for finding a vector that is orthogonal to two given vectors. By following the structured approach of setting up a matrix and applying the determinant method, one can efficiently compute the cross product and understand the geometric implications of the resulting vector.

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Problem

If vectors $v ⃗=⟨3,1,0⟩$ , $u ⃗=⟨0,-2,0⟩$ , and $w ⃗=v ⃗×u ⃗$ , find $w⃗⃗$ .

$w⃗=$\langle$0,0,-6$\rangle$$

$w⃗=$\langle$0,-2,0$\rangle$$

$w⃗=$\langle$0,0,6$\rangle$$

$w⃗=$\langle$0,0,-2$\rangle$$

Problem

If vectors $a⃗=5î$ , $b⃗=12k̂$ and $c⃗=a⃗$\times$ b⃗$ , find $c⃗$ .

$c⃗=60ĵ$

$c⃗=-60ĵ$

$c⃗=5î+12k̂$

$c⃗=60k̂$

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The cross product of two vectors is a vector that is perpendicular to both of the original vectors. It is calculated using a matrix with the unit vectors i, j, k in the first row, followed by the components of the vectors. The resulting vector is found by computing the determinant of this matrix. The cross product is useful in physics and engineering to find a vector orthogonal to a plane defined by two vectors.

To calculate the cross product of two vectors, follow these steps: 1) Set up a matrix with the unit vectors i, j, k in the first row. 2) Place the components of the first vector in the second row and the components of the second vector in the third row. 3) Compute the determinant of this matrix by multiplying and subtracting the appropriate elements. The resulting vector is the cross product, which can be expressed in component form.

In physics, the cross product is significant because it provides a way to find a vector that is perpendicular to two given vectors. This is particularly useful in applications such as calculating torque, where the direction of the torque vector is perpendicular to the plane formed by the force and the lever arm. The magnitude of the cross product also represents the area of the parallelogram formed by the two vectors.

The cross product has several important properties: 1) It is anti-commutative, meaning $a \times b = - b \times a$ . 2) It is distributive over vector addition, meaning $a \times (b + c) = a \times b + a \times c$ . 3) The cross product of parallel vectors is zero. 4) The magnitude of the cross product is equal to the area of the parallelogram formed by the two vectors.

The direction of the cross product vector is determined using the right-hand rule. Point the index finger of your right hand in the direction of the first vector (u) and your middle finger in the direction of the second vector (v). Your thumb will then point in the direction of the cross product (u × v). This ensures that the resulting vector is perpendicular to the plane formed by the original vectors.