Question

Given the two vectors \(\mathbf{b}=\hat{\mathbf{x}}+\hat{\mathbf{y}}\) and \(\mathbf{c}=\hat{\mathbf{x}}+\hat{\mathbf{z}}\) find \(\mathbf{b}+\mathbf{c}, 5 \mathbf{b}+2 \mathbf{c}, \mathbf{b} \cdot \mathbf{c},\) and \(\mathbf{b} \times \mathbf{c}\)

Short answer

\( \mathbf{b} + \mathbf{c} = (2, 1, 1), 5\mathbf{b} + 2\mathbf{c} = (7, 5, 2), \mathbf{b} \cdot \mathbf{c} = 1, \mathbf{b} \times \mathbf{c} = (1, 0, -1) \).

Step-by-step solution

Express the Vectors

First, express the vectors \( \mathbf{b} \) and \( \mathbf{c} \) in component form based on their definitions.\[ \mathbf{b} = \hat{\mathbf{x}} + \hat{\mathbf{y}} = (1, 1, 0) \] and \[ \mathbf{c} = \hat{\mathbf{x}} + \hat{\mathbf{z}} = (1, 0, 1) \]

Add the Vectors

To find \( \mathbf{b} + \mathbf{c} \), add the corresponding components of \( \mathbf{b} \) and \( \mathbf{c} \):\[ \mathbf{b} + \mathbf{c} = (1+1, 1+0, 0+1) = (2, 1, 1) \]

Scalar Multiplication and Addition

Find \( 5\mathbf{b} + 2\mathbf{c} \) by scaling each vector and then summing. Compute \( 5\mathbf{b} \): \[ 5\mathbf{b} = 5(1, 1, 0) = (5, 5, 0) \] Compute \( 2\mathbf{c} \): \[ 2\mathbf{c} = 2(1, 0, 1) = (2, 0, 2) \] Sum the results: \[ 5\mathbf{b} + 2\mathbf{c} = (5+2, 5+0, 0+2) = (7, 5, 2) \]

Dot Product

Calculate the dot product \( \mathbf{b} \cdot \mathbf{c} \). It is given by summing the products of the corresponding components: \[ \mathbf{b} \cdot \mathbf{c} = (1 \times 1) + (1 \times 0) + (0 \times 1) = 1 + 0 + 0 = 1 \]

Cross Product

Calculate the cross product \( \mathbf{b} \times \mathbf{c} \) using the determinant formula for \( \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{\mathbf{x}} & \hat{\mathbf{y}} & \hat{\mathbf{z}} \ 1 & 1 & 0 \ 1 & 0 & 1 \end{vmatrix} \). This results in: \[ \mathbf{b} \times \mathbf{c} = (1 \times 1 - 0 \times 0)\hat{\mathbf{y}} - (1 \times 1 - 0 \times 1)\hat{\mathbf{z}} + (1 \times 0 - 1 \times 1)\hat{\mathbf{x}} \] \[ = (1, 0, -1) \]

Key concepts

Vector Addition

Vector addition is a straightforward operation where we combine two vectors to form a new vector, usually referred to as the resultant vector. We perform vector addition by simply adding the corresponding components of the vectors involved.
In the exercise, to compute \( \mathbf{b} + \mathbf{c} \), you add each of the respective components of vectors \( \mathbf{b} = (1, 1, 0) \) and \( \mathbf{c} = (1, 0, 1) \).
Therefore, \( \mathbf{b} + \mathbf{c} \) results in a new vector:

• First component: \( 1 + 1 = 2 \)
• Second component: \( 1 + 0 = 1 \)
• Third component: \( 0 + 1 = 1 \)

Thus, the resultant vector is \( (2, 1, 1) \). By breaking down the addition into its individual components, we see precisely how new vectors are constructed by simple arithmetic.

Scalar Multiplication

Scalar multiplication involves stretching or shrinking a vector by multiplying it by a scalar, a real number. When multiplying a vector by a scalar, you scale each component of the vector by that number. This operation alters the magnitude of the vector without changing its direction, except when the scalar is negative, which reverses the direction.
To find \( 5\mathbf{b} \), you multiply each component of \( \mathbf{b} = (1, 1, 0) \) by 5:

• First component: \( 1 \times 5 = 5 \)
• Second component: \( 1 \times 5 = 5 \)
• Third component: \( 0 \times 5 = 0 \)

Therefore, \( 5\mathbf{b} = (5, 5, 0) \).
Next, for \( 2\mathbf{c} \), multiply each component of \( \mathbf{c} = (1, 0, 1) \) by 2:

• First component: \( 1 \times 2 = 2 \)
• Second component: \( 0 \times 2 = 0 \)
• Third component: \( 1 \times 2 = 2 \)

Hence, \( 2\mathbf{c} = (2, 0, 2) \). By adding these vectors, \( (5, 5, 0) + (2, 0, 2) \), the result is \( (7, 5, 2) \). Scalar multiplication simplifies into multiplying every vector component, leading to a new vector based on the chosen scalar.

Dot Product

The dot product is an operation that takes two vectors and returns a scalar. Unlike vector addition, which yields another vector, the dot product measures how much one vector extends in the direction of another. It is calculated by multiplying corresponding components of the two vectors and then summing the results.
For the vectors \( \mathbf{b} = (1, 1, 0) \) and \( \mathbf{c} = (1, 0, 1) \), the calculation proceeds as follows:

• Multiply the first components: \( 1 \times 1 = 1 \)
• Multiply the second components: \( 1 \times 0 = 0 \)
• Multiply the third components: \( 0 \times 1 = 0 \)

Sum the products: \( 1 + 0 + 0 = 1 \). Thus, the dot product \( \mathbf{b} \cdot \mathbf{c} = 1 \).
Here, the result is a measure of the degree to which two vectors point in the same direction, indicated by a single number or scalar.

Cross Product

The cross product of two vectors results in a new vector that is perpendicular to the plane formed by the original vectors, provided we work in three-dimensional space. It has applications in physics, particularly in scenarios involving rotational motion and torques. The magnitude and direction of the cross product depend on the angle and the orientation of the original vectors.
For \( \mathbf{b} = (1, 1, 0) \) and \( \mathbf{c} = (1, 0, 1) \), the calculation is based on the determinant of a matrix formed by the unit vectors \( \hat{\mathbf{x}}, \hat{\mathbf{y}}, \hat{\mathbf{z}} \) and the components of the vectors. The operation can be broken down into these components:

• \( ( \mathbf{b}_y \times \mathbf{c}_z - \mathbf{b}_z \times \mathbf{c}_y ) \hat{\mathbf{x}} = (1 \times 1 - 0 \times 0) \hat{\mathbf{x}} = 1 \hat{\mathbf{x}} \)
• \( -( \mathbf{b}_x \times \mathbf{c}_z - \mathbf{b}_z \times \mathbf{c}_x ) \hat{\mathbf{y}} = -(1 \times 1 - 0 \times 1) \hat{\mathbf{y}} = -1 \hat{\mathbf{y}} \)
• \( ( \mathbf{b}_x \times \mathbf{c}_y - \mathbf{b}_y \times \mathbf{c}_x ) \hat{\mathbf{z}} = (1 \times 0 - 1 \times 1) \hat{\mathbf{z}} = -1 \hat{\mathbf{z}} \)

This results in the vector \( \mathbf{b} \times \mathbf{c} = (1, -1, -1) \), representing a new vector that is orthogonal to both \( \mathbf{b} \) and \( \mathbf{c} \), making it a central tool in vector physics and engineering.