Verify the Cauchy-Schwarz inequality for each of the...

Verify the Cauchy-Schwarz inequality for each of the following pairs of vectors $\mathbf{v}, \mathbf{w}$, using the standard dot product, and then determine the angle between them: (a) $(1,2)^T,(-1,2)^T$, (b) $(1,-1,0)^T,(-1,0,1)^T$, (c) $(1,-1,0)^T,(2,2,2)^T$, (d) $(1,-1,1,0)^T,(-2,0,-1,1)^T$, (e) $(2,1,-2,-1)^T,(0,-1,2,-1)^T$.

Verify the Cauchy-Schwarz inequality for each of the following pairs of vectors $\mathbf{v}, \mathbf{w}$, using the standard dot product, and then determine the angle between them:
(a) $(1,2)^T,(-1,2)^T$,
(b) $(1,-1,0)^T,(-1,0,1)^T$,
(c) $(1,-1,0)^T,(2,2,2)^T$,
(d) $(1,-1,1,0)^T,(-2,0,-1,1)^T$,
(e) $(2,1,-2,-1)^T,(0,-1,2,-1)^T$.

Key Concepts

Dot Product

The dot product is a fundamental operation in Euclidean space that multiplies corresponding elements of two vectors and sums the results. It serves as a measure of the similarity between the directions of the two vectors and plays a central role in defining both vector magnitude and the angle between vectors.

Euclidean Norm

The Euclidean norm, or length, of a vector is calculated as the square root of the sum of the squares of its components. It is crucial for measuring the size of a vector, and it is used in conjunction with the dot product to normalize vectors and compute angles between them.

Cauchy-Schwarz Inequality

The Cauchy-Schwarz inequality is a key result in linear algebra and analysis that asserts the absolute value of the dot product of two vectors is less than or equal to the product of their Euclidean norms. This inequality assures that the cosine of the angle between two vectors, derived from the dot product formula, will always lie between -1 and 1.

Angle Between Vectors

The angle between two vectors can be determined using the formula cos(?) = (v · w) / (||v|| ||w||), where v · w is the dot product and ||v|| and ||w|| are the Euclidean norms of the vectors. This relationship not only provides a measure of directional difference but also plays a vital role in applications such as projections and orthogonality.

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