91Ó°ÊÓ

The Cauchy-Schwarz inequality is a fundamental result in vector calculus. It states that for any vectors \(\mathbf{V}\) and \(\mathbf{W}\) in an inner product space, the absolute value of their dot product is less than or equal to the product of their magnitudes: \[|\mathbf{V} \cdot \mathbf{W}| \leq \|\mathbf{V}\| \|\mathbf{W}\|\]This inequality is pivotal when analyzing vector magnitudes because it assures us that the dot product's contribution cannot excessively inflate compared to individual vector lengths. This is particularly important in exercise **(a)**, where we explored the situation \(|\mathbf{V} + \mathbf{W}|^2\), which simplifies to \(|\mathbf{V}|^2 + |\mathbf{W}|^2 + 2\mathbf{V} \cdot \mathbf{W}|\). The Cauchy-Schwarz inequality suggests that - The term \(2\mathbf{V} \cdot \mathbf{W}\) is maximized when vectors are parallel. - If \(\mathbf{V}\) and \(\mathbf{W}\) are not orthogonal, this term can influence the sum significantly.This makes the statement false if the vectors aren't perpendicular.

The dot product, also known as the scalar product, is an essential operation in vector algebra. It combines two vector quantities to produce a scalar, calculated as:\[\mathbf{V} \cdot \mathbf{W} = \|\mathbf{V}\|\|\mathbf{W}\|\cos \theta\]where \(\theta\) is the angle between \(\mathbf{V}\) and \(\mathbf{W}\). It offers valuable insights into vector relationships:

• Magnitude: If the dot product is zero, vectors are orthogonal.
• Direction: A positive dot product implies vectors pointing in the same direction, while a negative value implies opposite directions.

For exercise **(c)**, this principle is invaluable because the operation is commutative, meaning \(\mathbf{V} \cdot \mathbf{W} = \mathbf{W} \cdot \mathbf{V}\). This property simplifies many calculations in physics and engineering, highlighting how vectors interact through their orientations.

The triangle inequality is a principle stating that for any vectors \(\mathbf{V}\) and \(\mathbf{W}\), the magnitude of their sum is no greater than the sum of their magnitudes:\[|\mathbf{V} + \mathbf{W}| \leq |\mathbf{V}| + |\mathbf{W}|\]This is a geometric interpretation of how vector addition works:

• Equality holds only when the vectors are parallel or collinear.
• If vectors form an acute angle, their sum is less than the total sum of magnitudes.

In exercise **(b)**, this principle explains potential equality. However, for real triangles, this equality holds true only if the direction of vectors matches, leading the statement given to be false if they aren't precisely collinear.

Orthogonality describes a situation where two vectors, \(\mathbf{A}\) and \(\mathbf{B}\), are perpendicular. The key condition is that their dot product must be zero:\[\mathbf{A} \cdot \mathbf{B} = 0\]This concept emerges frequently in mathematics and physics. When dealing with dimensions higher than two, vectors cannot simply be 'straight lines' like in familiar 2D geometry. They must fulfill the perpendicular condition, forming a plane:

• Other vectors perpendicular to a certain vector form a set that lies on a plane.
• The concept simplifies solving equations involving perpendicularity.

In exercise **(d)**, vectors perpendicular to \(\mathrm{i} + \mathrm{j} + \mathrm{k}\) do not align along a line but cover a plane, as their direction isn't confined to one principal axis alone, making the statement false.