Show that the vectors are not orthogonal with respect to the...

Show that the vectors are not orthogonal with respect to the Euclidean inner product on $R^{2}$, and then find a value of $k$ for which the vectors are orthogonal with respect to the weighted Euclidean inner product $(\mathbf{u}, \mathbf{v})=2 u_{1} v_{1}+k u_{2} v_{2}$. $$\mathbf{u}=(2,-4), \mathbf{v}=(0,3)$$

Key Concepts

Euclidean Inner Product

The Euclidean inner product (or dot product) in R² is defined as the sum of the products of the corresponding components of two vectors. It is a standard way to measure angles and lengths in Euclidean space, and its fundamental property is that it allows us to determine the angle between two vectors or check for orthogonality by evaluating if the result is zero.

Orthogonality

Two vectors are said to be orthogonal if their inner product is zero; this indicates that they are perpendicular to each other in the geometric sense. In an inner product space, checking the value of the inner product provides a direct test for whether the vectors meet at a right angle.

Weighted Euclidean Inner Product

The weighted Euclidean inner product modifies the standard inner product by incorporating weight factors for different coordinates. In this generalized inner product, each product of corresponding components is multiplied by a specified weight, which adjusts the contribution of each component in determining the overall measure of angle and length.

Parameter Adjustment for Orthogonality

Adjusting a parameter within a weighted inner product setup provides a way to influence the condition for orthogonality between vectors. By setting the modified inner product equal to zero and solving for the parameter, one can find the specific conditions or weight values that cause the orthogonality, demonstrating how inner product definitions can be tailored to yield desired geometric properties.

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