Question

Two unit vectors, \(\mathbf{a}_{1}\) and \(\mathbf{a}_{2}\), lie in the \(x y\) plane and pass through the origin. They make angles \(\phi_{1}\) and \(\phi_{2}\), respectively, with the \(x\) axis \((a)\) Express each vector in rectangular components; \((b)\) take the dot product and verify the trigonometric identity, \(\cos \left(\phi_{1}-\phi_{2}\right)=\cos \phi_{1} \cos \phi_{2}+\sin \phi_{1} \sin \phi_{2} ;(c)\) take the cross product and verify the trigonometric identity \(\sin \left(\phi_{2}-\phi_{1}\right)=\sin \phi_{2} \cos \phi_{1}-\cos \phi_{2} \sin \phi_{1}\)

Short answer

Question: Verify the trigonometric identities involving the dot and cross products of two unit vectors \(\mathbf{a}_{1}\) and \(\mathbf{a}_{2}\) lying in the xy-plane with angles \(\phi_{1}\) and \(\phi_{2}\) measured counterclockwise from the positive x-axis. Answer: The first trigonometric identity, \(\cos(\phi_{1} -\phi_{2}) = \cos\phi_{1}\cos\phi_{2} + \sin\phi_{1}\sin\phi_{2}\), corresponds to the dot product of the two unit vectors, \(\mathbf{a_{1}}\cdot\mathbf{a_{2}}\). The second trigonometric identity, \(\sin(\phi_{2} - \phi_{1}) = \sin\phi_{2}\cos\phi_{1} - \cos\phi_{2}\sin\phi_{1}\), corresponds to the cross product of the two unit vectors, \(\mathbf{a_{1}}\times\mathbf{a_{2}}\). Both identities are verified using the angle subtraction formulas for sine and cosine.

Step-by-step solution

Express the vectors in rectangular components

To find the rectangular components of a unit vector, we can use the formulas \(\mathbf{a_{1}} = \langle\cos\phi_{1}, \sin\phi_{1}\rangle\) and \(\mathbf{a_{2}} = \langle\cos\phi_{2}, \sin\phi_{2}\rangle\), since the magnitudes of the unit vectors are \(1\). These components are derived from the definitions of sine and cosine.

Compute the dot product

The dot product of two vectors \(\mathbf{a_{1}}\) and \(\mathbf{a_{2}}\) is given by the formula \(\mathbf{a_{1}}\cdot\mathbf{a_{2}} = a_{1x} a_{2x} + a_{1y} a_{2y}\). Substituting the rectangular components, we get: $$\mathbf{a_{1}}\cdot\mathbf{a_{2}} = (\cos\phi_{1})(\cos\phi_{2}) + (\sin\phi_{1})(\sin\phi_{2})$$

Verify the first trigonometric identity

Now, we will verify the identity \(\cos(\phi_{1} -\phi_{2}) = \cos\phi_{1}\cos\phi_{2} + \sin\phi_{1}\sin\phi_{2}\). By using the angle subtraction formula for cosine, we get: $$\cos(\phi_{1} - \phi_{2}) = \cos\phi_{1}\cos\phi_{2} + \sin\phi_{1}\sin\phi_{2}$$ Comparing this with the expression for the dot product in Step 2, we can confirm that the first trigonometric identity holds.

Compute the cross product

The cross product of two vectors in the \(xy\) plane is given by the formula \(\mathbf{a_{1}}\times\mathbf{a_{2}} = (a_{1x}a_{2y} - a_{1y}a_{2x})\mathbf{\hat{z}}\). Substituting the rectangular components, we get: $$\mathbf{a_{1}}\times\mathbf{a_{2}} = [(\cos\phi_{1})(\sin\phi_{2}) - (\sin\phi_{1})(\cos\phi_{2})]\mathbf{\hat{z}}$$

Verify the second trigonometric identity

Now, we will verify the identity \(\sin(\phi_{2} - \phi_{1}) = \sin\phi_{2}\cos\phi_{1} - \cos\phi_{2}\sin\phi_{1}\). By using the angle subtraction formula for sine, we get: $$\sin(\phi_{2} - \phi_{1}) = \sin\phi_{2}\cos\phi_{1} - \cos\phi_{2}\sin\phi_{1}$$ Comparing this with the expression for the cross product in Step 4, we can confirm that the second trigonometric identity holds.

Key concepts

Rectangular Components

When expressing vectors in the xy-plane, rectangular components provide a clear way to visualize them using perpendicular axes, usually the x-axis and y-axis. Each vector

• has an x-component, which is the projection of the vector on the x-axis.
• has a y-component, which is the projection on the y-axis.

For unit vectors, which have a length of 1, these components relate directly to the angles they make with the x-axis. If a vector makes an angle \( \phi \) with the x-axis, its rectangular components are:

• x-component: \( \cos(\phi) \)
• y-component: \( \sin(\phi) \)

So, the vector \( \mathbf{a} = \langle \cos(\phi), \sin(\phi) \rangle \). This representation helps in performing vector operations like addition, subtraction, and finding dot and cross products.

Dot Product

The dot product, or scalar product, is a way to multiply two vectors, producing a scalar. It measures how much one vector extends in the direction of another. The formula for the dot product is:

• \( \mathbf{a} \cdot \mathbf{b} = a_x b_x + a_y b_y \)

Using the unit vectors discussed, \( \mathbf{a_1} = \langle \cos(\phi_1), \sin(\phi_1) \rangle \) and \( \mathbf{a_2} = \langle \cos(\phi_2), \sin(\phi_2) \rangle \), the dot product becomes:

• \( \cos(\phi_1)\cos(\phi_2) + \sin(\phi_1)\sin(\phi_2) \)

One key property is its relation to the cosine of the angle difference: \( \cos(\phi_1 - \phi_2) = \cos(\phi_1)\cos(\phi_2) + \sin(\phi_1)\sin(\phi_2) \). This shows the alignment of the vectors in terms of their angle difference.

Cross Product

The cross product is another way to combine two vectors, but it results in a vector perpendicular to both. In the 2D xy-plane, the result is along the z-axis. The cross product magnitude is given by:

• \( \mathbf{a_1} \times \mathbf{a_2} = (a_{1x}a_{2y} - a_{1y}a_{2x}) \mathbf{\hat{z}} \)

For our unit vectors:

• \( \mathbf{a_1} \times \mathbf{a_2} = (\cos(\phi_1)\sin(\phi_2) - \sin(\phi_1)\cos(\phi_2)) \mathbf{\hat{z}} \)

This aligns with the identity \( \sin(\phi_2 - \phi_1) = \sin(\phi_2)\cos(\phi_1) - \cos(\phi_2)\sin(\phi_1) \). The cross product thus captures the sine of the angle difference, showing the orientation difference.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the involved variables. They are crucial in connecting angles and side lengths in triangles and can be applied in vector mathematics:

• The cosine difference identity is \( \cos(\phi_1 - \phi_2) = \cos(\phi_1)\cos(\phi_2) + \sin(\phi_1)\sin(\phi_2) \), closely tied with the dot product.
• The sine difference identity is \( \sin(\phi_2 - \phi_1) = \sin(\phi_2)\cos(\phi_1) - \cos(\phi_2)\sin(\phi_1) \), reflecting in the cross product.

These identities help unravel vector relationships and simplify complex expressions, making vector operations more intuitive.