problem04

4. Cross product: geometry vs components.

1. The geometric definition of crossproduct is this \(\vec{a} \times \vec{b}\) is a vector \(\vec{c}\) with magnitude \(\|\vec{a}\|\|\vec{b}\|\sin\theta_{ab}\) that is orthmorogonal to \(\vec{a}\) and \(vec{b}\) in the direction given by the right hand rule. Use this definition to find an alternative geometric definition involving projection (namely: project \(\vec{b}\) onto theplane that is orthogonal to \(\vec{a}\); then stretch it by \(\vec{|a|}\); then rotate it \(\pi/2\) around the \(\vec{a}\) axis). Use that definition to show the distributive rule \(\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}\).
   
2. Then use the distributive rule to find the component formula for cross product,namely that \(\vec{a} \times \vec{b} = (a_2 b_3 − a_3 b_2)\hat{e}_1 + (a_3 b_1 − a_1 b_3) \hat{e}_2 + (a_1 b_2 − a_2 b_1)\hat{e}_3\).

Note that, this distributive law implies that, for given \(va \times \vec{v}\) is a linear operator.That is \(va \times \vec{v}\) is a linear function fo \(\vec{v}\). Later in the course we will use this to replacethe cross product with a tensor product. Hint: You can read about this in, say, theRuina/Pratap book (box 1.7).

1 The geometric definition of crossproduct is this \(\vec{a} \times \vec{b}\) is a vector \(\vec{c}\) with magnitude \(\|\vec{a}\|\|\vec{b}\|\sin\theta_{ab}\) that is orthmorogonal to \(\vec{a}\) and \(vec{b}\) in the direction given by the right hand rule. Use this definition to find an alternative geometric definition involving projection (namely: project \(\vec{b}\) onto theplane that is orthogonal to \(\vec{a}\); then stretch it by \(\vec{|a|}\); then rotate it \(\pi/2\) around the \(\vec{a}\) axis). Use that definition to show the distributive rule \(\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}\).

The cross product \(\vec{a} \times \vec{b}\) is defined geometrically as a vector \(\vec{c}\) with magnitude \(\|\vec{a}\|\|\vec{b}\|\sin \theta_{ab}\), direction orthogonal to both \(\vec{a}\) and \(\vec{b}\), and orientation following the right-hand rule. We can reformulate this using projection. First, project \(\vec{b}\) onto the plane orthogonal to \(\vec{a}\) using the formula \(\vec{b}_{\perp} = \vec{b} - (\vec{b}\cdot\hat{a})\hat{a}\), where \(\hat{a}\) is the unit vector in direction of \(\vec{a}\). The magnitude of this projection is \(\|\vec{b}_{\perp}\| = \|\vec{b}\|\sin \theta\)

Then stretch this vector by \(\|\vec{a}\|\), giving \(\|\vec{a}\|\vec{b}_{\perp}\) with magnitude \(\|\vec{a}\|\|\vec{b}\|\sin \theta\). Finally, rotate this vector by \(\pi/2\) around the \(\vec{a}\) axis, which preserves magnitude while making the result orthogonal to both \(\vec{a}\) and \(\vec{b}_{\perp}\) in the right-hand rule direction. This construction yields a vector identical to the original definition.

2 Then use the distributive rule to find the component formula for cross product,namely that \(\vec{a} \times \vec{b} = (a_2 b_3 − a_3 b_2)\hat{e}_1 + (a_3 b_1 − a_1 b_3) \hat{e}_2 + (a_1 b_2 − a_2 b_1)\hat{e}_3\).

To prove distributivity, \(\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}\), we use this new geometric definition. When we project \((\vec{b} + \vec{c})\) onto the plane perpendicular to \(\vec{a}\), linearity of projection gives us \((\vec{b} + \vec{c})_{\perp} = \vec{b}_{\perp} + \vec{c}_{\perp}\). Stretching by \(\|\vec{a}\|\) is also linear: \(\|\vec{a}\|(\vec{b}_{\perp} + \vec{c}_{\perp}) = \|\vec{a}\|\vec{b}_{\perp} + \|\vec{a}\|\vec{c}_{\perp}\). Since rotation by \(\pi/2\) is linear, \(R_{\pi/2}(\vec{b}_{\perp} + \vec{c}_{\perp}) = R_{\pi/2}\vec{b}_{\perp} + R_{\pi/2}\vec{c}_{\perp}\), proving the distributive property.

To derive the component formula, we use distributivity: \(\vec{a} \times \vec{b} = (a_1\hat{e}_1 + a_2\hat{e}_2 + a_3\hat{e}_3) \times (b_1\hat{e}_1 + b_2\hat{e}_2 + b_3\hat{e}_3) = \sum_{i,j} a_ib_j(\hat{e}_i \times \hat{e}_j)\). Using the standard basis cross products (\(\hat{e}_1 \times \hat{e}_2 = \hat{e}_3\), \(\hat{e}_2 \times \hat{e}_3 = \hat{e}_1\), \(\hat{e}_3 \times \hat{e}_1 = \hat{e}_2\), \(\hat{e}_i \times \hat{e}_i = 0\), \(\hat{e}_j \times \hat{e}_i = -(\hat{e}_i \times \hat{e}_j)\)), we expand and group terms to get the final component formula: \(\vec{a} \times \vec{b} = (a_2b_3 - a_3b_2)\hat{e}_1 + (a_3b_1 - a_1b_3)\hat{e}_2 + (a_1b_2 - a_2b_1)\hat{e}_3\).