Question

Prove or disprove For fixed values of \(a, b, c,\) and \(d,\) the value of proj \(_{(k a, k b)}\langle c, d\rangle\) is constant for all nonzero values of \(k,\) for \(\langle a, b\rangle \neq\langle 0,0\rangle\).

Short answer

Based on the step-by-step solution, the projection of vector $\vec{a} = \langle c, d \rangle$ onto vector $\vec{b} = \langle k a, k b \rangle$ was calculated to be: $$\text{proj}_{\vec{b}} \vec{a} = \frac{c a + d b}{k(a^2 + b^2)} \langle k a, k b \rangle$$ Since the projection vector is dependent on \(k\), the value of the projection is not constant for all nonzero values of \(k\). Thus, the statement is disproved.

Step-by-step solution

Write down the projection formula for the given vectors

We are given the vectors: \(\vec{a} = \langle c, d \rangle\) and \(\vec{b} = \langle k a, k b \rangle\). Write down the projection formula for these vectors: $$\text{proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{\|\vec{b}\|^2} \vec{b}$$

Compute the dot product of \(\vec{a}\) and \(\vec{b}\)

Compute the dot product: $$\vec{a} \cdot \vec{b} = \langle c, d \rangle \cdot \langle k a, k b \rangle = ck a + dk b$$

Compute the norm squared of \(\vec{b}\)

Compute the square of the norm of \(\vec{b}:\) $$\|\vec{b}\|^2 = \langle k a, k b \rangle \cdot \langle k a, k b \rangle = (k a)^2 + (k b)^2 = k^2(a^2 + b^2)$$

Calculate the projection

Using the values found in steps 2 and 3, calculate the projection: $$\text{proj}_{\vec{b}} \vec{a} = \frac{ck a + dk b}{k^2(a^2 + b^2)} \langle k a, k b \rangle = \frac{ck a + dk b}{k^2(a^2 + b^2)} \langle k a, k b \rangle$$

Simplify the expression

Divide both terms in the numerator by \(k\): $$\text{proj}_{\vec{b}} \vec{a} = \frac{c a + d b}{k(a^2 + b^2)} \langle k a, k b \rangle$$ It can be observed that the projection vector is dependent on \(k\), so the value of the projection is not constant for all nonzero values of \(k\). Thus, the statement is disproved.

Key concepts

Dot Product

The dot product is a fundamental operation in vector algebra. It allows us to multiply two vectors, resulting in a scalar quantity. This scalar provides vital information about the relationship between the two vectors. Consider two vectors, \(\vec{u} = \langle u_1, u_2 \rangle\) and \(\vec{v} = \langle v_1, v_2 \rangle\). The dot product is calculated as:
\[ \vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 \]
This operation tells us about the angle between the vectors:

• If the dot product is positive, the angle is acute.
• If the dot product is zero, the vectors are perpendicular.
• If the dot product is negative, the angle is obtuse.

Understanding the dot product is crucial in evaluating vector projections and understanding their significance in geometry and physics.

Vector Norm

The vector norm, often referred to as the "magnitude" or "length" of a vector, is an important concept in vector calculus. It gives us a measure of how long the vector is. For a vector \(\vec{v} = \langle v_1, v_2 \rangle\), the norm is calculated as follows:
\[ \|\vec{v}\| = \sqrt{v_1^2 + v_2^2} \]
In practical terms, the norm is used to gauge the extent or size of the vector in its space:

• The norm is always a non-negative number, being zero only for the zero vector.
• It is invariant under rotation and simply scales linearly with the scaling of the vector.
• The square of the norm appears in calculating the projection of one vector onto another.

A strong grasp of vector norms is essential for mastering concepts involving projections and various vector operations.

Projection Formula

The projection formula is a fantastic tool to cast one vector onto another. This process effectively generates a new vector that lies along the direction of another vector. For two vectors, \(\vec{a}\) and \(\vec{b}\), the projection of \(\vec{a}\) onto \(\vec{b}\) is given by:
\[ \text{proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{\|\vec{b}\|^2} \vec{b} \]
Here is how this formula plays out:

• The numerator, \(\vec{a} \cdot \vec{b}\), gives the component of \(\vec{a}\) that aligns with \(\vec{b}\).
• The denominator, \(\|\vec{b}\|^2\), scales the projection correctly along \(\vec{b}\).
• The resulting vector aligns with \(\vec{b}\) and represents \(\vec{a}\)'s effect in the direction of \(\vec{b}\).

Mastering the projection formula helps in areas like physics, where understanding forces along different axes is crucial. In the exercise given, it is shown that the projection is not constant over all scalings, emphasizing the importance of checking dependencies on all parameters.