Question

If \(\vec{a}\) and \(\vec{b}\) are vectors in space given by \(\vec{a}=\frac{\hat{i}-2 \hat{j}}{\sqrt{5}}\) and \(\vec{b}=\frac{2 \hat{i}+\hat{j}+3 \hat{k}}{\sqrt{14}}\), then the value of \((2 \vec{a}+\vec{b}) \cdot[(\vec{a} \times \vec{b}) \times(\vec{a}-2 \vec{b})]\) is

Short answer

Perform a series of vector operations including cross products, dot products, and scalar multiplications, and simplify using properties of vector products to obtain the final numerical value.

Step-by-step solution

- Simplify the scalar triple product

The value inside the brackets is a scalar triple product which can be simplified using the fact that \(\vec{A} \cdot (\vec{B} \times \vec{C}) = \vec{C} \cdot (\vec{A} \times \vec{B})\). So, we get \(\vec{a} \times \vec{b}) \times(\vec{a}-2\vec{b}) = (\vec{a}-2\vec{b}) \cdot (\vec{a} \times \vec{b})\).

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

To find the cross product, use the determinant of a matrix with the unit vectors as the first row and the respective components of \(\vec{a}\) and \(\vec{b}\) as the second and third rows. The result will give us the vector that is orthogonal to both \(\vec{a}\) and \(\vec{b}\).

- Perform the vector subtraction and scalar multiplication

We need to calculate \(\vec{a} - 2\vec{b}\) by performing the appropriate vector subtraction and scalar multiplication, which involves multiplying each component of \(\vec{b}\) by 2 and then subtracting from the respective components of \(\vec{a}\).

- Compute the dot product

Use the results from the previous steps to compute the dot product of the vector \(\vec{a} - 2\vec{b}\) and the cross product \(\vec{a} \times \vec{b}\). This involves multiplying the respective components and summing them.

- Calculate the vector addition and scalar multiplication of \(2\vec{a}+\vec{b}\)

Calculate \(2\vec{a}+\vec{b}\) by multiplying each component of \(\vec{a}\) by 2, then adding the respective components of \(\vec{b}\).

- Compute the final dot product

Take the dot product of the vector from Step 5, \(2\vec{a}+\vec{b}\), with the result from Step 4 to get the final answer.

Key concepts

Scalar Triple Product

The scalar triple product of vectors involves three vectors and produces a scalar value. It is defined as the dot product of one vector with the cross product of the other two. The notation is \( \vec{A} \cdot (\vec{B} \times \vec{C}) \). This product gives the volume of the parallelepiped formed by the vectors and thus has geometric significance.

When computing the scalar triple product, order matters due to the cross product's directionality; however, the cyclical order doesn't change the value, explaining the property that \( \vec{A} \cdot (\vec{B} \times \vec{C}) = \vec{C} \cdot (\vec{A} \times \vec{B}) \). If the volume calculated is zero, it indicates that the vectors are coplanar.

Vector Cross Product

The cross product, denoted \( \vec{A} \times \vec{B} \), results in a vector that is orthogonal (perpendicular) to the plane containing \( \vec{A} \) and \( \vec{B} \). Its magnitude represents the area of the parallelogram with sides \( \vec{A} \) and \( \vec{B} \).

This operation obeys the right-hand rule, which determines the direction of the resulting vector. Calculating the cross product typically involves setting up a determinant using the unit vectors \(\textbf{i}\), \(\textbf{j}\), and \(\textbf{k}\) in the first row and the components of \( \vec{A} \) and \( \vec{B} \) below them, which aids in memorizing the process and ensures accuracy.

Vector Dot Product

Also referred to as the inner product or scalar product, the vector dot product, symbolized as \( \vec{A} \cdot \vec{B} \), combines two vectors to yield a scalar. It is calculated by multiplying corresponding components of the vectors and summing the products.

The dot product is significant as it also indicates the cosine of the angle between the two vectors. If the dot product is zero, the vectors are perpendicular, and if it is positive or negative, it tells the angle is acute or obtuse, respectively. This characteristic is critical when analyzing vector orientations in various applications.

Vector Subtraction

Vector subtraction involves finding the difference between two vectors, leading to a new vector. It is denoted by \( \vec{A} - \vec{B} \). Subtracting vectors graphically can be visualized by flipping \( \vec{B} \) directionally and adding it to \( \vec{A} \). Analytically, it's done by subtracting each component of \( \vec{B} \) from the corresponding component of \( \vec{A} \).

The result is a vector that denotes the displacement from the end of \( \vec{B} \) to the end of \( \vec{A} \). Understanding the process of vector subtraction is crucial for resolving forces, calculating moment arms, and more in physics and engineering.

Scalar Multiplication of Vectors

Scalar multiplication involves multiplying a vector by a scalar value, altering the magnitude of the vector but not its direction, unless the scalar is negative, in which case the direction is reversed. This operation is written as \( c\vec{A} \), where \( c \) is a scalar and \( \vec{A} \) is a vector.

Each component of the vector is multiplied by the scalar, effectively stretching or compressing the vector's length. Scalar multiplication is integral to operations such as vector scaling, projection, and in representing physical concepts like velocity, where the scalar signifies the speed, and the vector provides the direction.