91Ó°ÊÓ

In mathematics and physics, vector subtraction is an essential operation used to determine the difference between two vectors. It involves calculating the difference between corresponding components of each vector. Consider two vectors \( \mathbf{u} = \langle a_1, b_1, c_1 \rangle \) and \( \mathbf{v} = \langle a_2, b_2, c_2 \rangle \). The vector subtraction \( \mathbf{u} - \mathbf{v} \) is calculated as follows:

• Subtract the corresponding x-components: \( a_1 - a_2 \)
• Subtract the corresponding y-components: \( b_1 - b_2 \)
• Subtract the corresponding z-components: \( c_1 - c_2 \)

This results in a new vector: \( \mathbf{u} - \mathbf{v} = \langle a_1 - a_2, b_1 - b_2, c_1 - c_2 \rangle \). In the original exercise, the vector subtraction yielded the vector \( \langle 2 \cos t, -2 \sin t, 0 \rangle \). Notice that the components reflect the directional difference between vectors \( \mathbf{u} \) and \( \mathbf{v} \). It shows how far and in which direction \( \mathbf{u} \) is from \( \mathbf{v} \) when moved along the coordinate axes, providing a clear geometric interpretation.

Scalar multiplication is an operation where each component of a vector is multiplied by a scalar—a real number. This process changes the vector's magnitude but not its direction (unless the scalar is negative, which reverses the direction). Suppose you have a vector \( \mathbf{u} = \langle a, b, c \rangle \) and a scalar \( k \). The result of scalar multiplication \( k \mathbf{u} \) is given by:

• Multiply the x-component: \( ka \)
• Multiply the y-component: \( kb \)
• Multiply the z-component: \( kc \)

This leads to a new vector: \( k \mathbf{u} = \langle ka, kb, kc \rangle \). For example, multiplying the vector \( \mathbf{u} = \langle 2 \cos t, -2 \sin t, 3 \rangle \) by \(-2\) yields \( \langle -4 \cos t, 4 \sin t, -6 \rangle \). Here, the negative scalar changes the direction of the vector, effectively flipping it while also scaling its size by a factor of 2. This demonstrates scalar multiplication's ability to both change the magnitude and alter the orientation when the scalar is negative.

The Pythagorean identity is a fundamental trigonometric formula that states: \[ \cos^2 t + \sin^2 t = 1 \]This identity asserts that for any angle \( t \), the sum of the squares of its cosine and sine is always equal to one. It's hugely significant in vector mathematics, especially in calculating vector magnitudes. For any vector \( \langle a, b, 0 \rangle \), the magnitude is calculated using the formula \( \sqrt{a^2 + b^2 + c^2} \). When the components \( a \) and \( b \) are expressed with trigonometric functions, the Pythagorean identity simplifies calculations:

• Replace \( a^2 + b^2 \) with \( (\cos^2 t + \sin^2 t) \), which is 1.
• Apply the identity to derive a cleaner result.

In the provided solution, the application of the Pythagorean identity simplified the vector magnitude \( \sqrt{4 (\cos^2 t + \sin^2 t)} = \sqrt{4} = 2 \). This makes it easier to calculate magnitudes when dealing with trigonometric expressions by reducing the complexity of the algebra involved.