Question

Let \(\vec{a}=-\hat{i}-\hat{k}, \vec{b}=-\hat{i}+\hat{j}\) and \(\vec{c}=\hat{i}+2 \hat{j}+3 \hat{k}\) be three given vectors. If \(\vec{r}\) is a vector such that \(\vec{r} \times \vec{b}=\vec{c} \times \vec{b}\) and \(\vec{r} \cdot \vec{a}=0\), then the value of \(\vec{r} \cdot \vec{b}\) is

Short answer

\(\vec{r} \cdot \vec{b} = 0\) since the component of \(\vec{r}\) along \(\vec{a}\) is zero and \(\vec{a}\) is not parallel to \(\vec{b}\).

Step-by-step solution

Calculate the cross product \(\vec{c} \times \vec{b}\)

Using the cross product formula for vectors, calculate the cross product of \(\vec{c}\) and \(\vec{b}\). This step is necessary to find a vector that is perpendicular to both \(\vec{c}\) and \(\vec{b}\).

Equate \(\vec{r} \times \vec{b}\) to \(\vec{c} \times \vec{b}\)

Since \(\vec{r} \times \vec{b}\) equals \(\vec{c} \times \vec{b}\), set the results from the previous step equal to \(\vec{r} \times \vec{b}\). This equation will provide components of \(\vec{r}\).

Find \(\vec{r}\) using \(\vec{r} \cdot \vec{a} = 0\)

Knowing that \(\vec{r}\) is orthogonal to \(\vec{a}\) because their dot product is zero, find the components of \(\vec{r}\) that satisfy both the cross product equation and the dot product condition.

Calculate the dot product \(\vec{r} \cdot \vec{b}\)

Once the vector \(\vec{r}\) is found, calculate the dot product of \(\vec{r}\) and \(\vec{b}\) to obtain the value of \(\vec{r} \cdot \vec{b}\).

Key concepts

Vector Algebra

Understanding the basic principles of vector algebra is crucial when solving physics and engineering problems. A vector is a mathematical entity that has both magnitude and direction, making it different from a scalar, which has only magnitude. Simple operations include vector addition and subtraction, where vectors can be combined head-to-tail or subtracted by reversing the direction of one and then adding.

Multiplying a vector by a scalar scales the vector in size without changing its direction. These operations can be visualized graphically, or by using algebraic representations, where vectors are often written in terms of their components along the coordinate axes, using unit vectors like \( \hat{i}, \hat{j}, \hat{k} \) for the x, y, and z directions respectively. For example, the vector \( \vec{a} = -\hat{i}-\hat{k} \) has components of -1 along the x-axis and -1 along the z-axis with no y-component.

Cross Product Calculation

The cross product, also known as the vector product, is a binary operation on two vectors in three-dimensional space. It has the essential property of being perpendicular to both of the original vectors, creating a new vector that is fundamentally linked to the orientation of the original pair.

To calculate the cross product of two vectors, we employ a determinant-like method using the unit vectors \( \hat{i}, \hat{j}, \hat{k} \) as the 'template'. The components of the vectors are filled in such that the x-components are multiplied by \( \hat{i} \) and so forth. The result is a new vector, for example, \( \vec{c} \times \vec{b} \) in the provided exercise. Remember that the cross-product is anticommutative, meaning changing the order of the vectors changes the sign of the result.

Dot Product

The dot product or scalar product is another way to combine two vectors, yielding a scalar result. Key in understanding the dot product is that it reflects the degree of alignment between the two vectors. If they are perfectly aligned, the dot product is maximized, equaling the product of the magnitudes of the two vectors. Conversely, if the vectors are orthogonal (at a 90-degree angle), their dot product is zero.

Calculating the dot product algebraically involves multiplying the corresponding components of each vector and summing the results. For instance, \( \vec{r} \cdot \vec{a} = 0 \) suggests that \( \vec{r} \) is orthogonal to \( \vec{a} \) as presented in the exercise. The general formula for the dot product in terms of components is \( \vec{r} \cdot \vec{a} = r_x a_x + r_y a_y + r_z a_z \) where \( r_x, r_y, r_z \) and \( a_x, a_y, a_z \) are the components of vectors \( \vec{r} \) and \( \vec{a} \) respectively.