Question

if \(\underline{a}+m \underline{b}+3 \underline{c},-2 \underline{a}+3 \underline{b}-4 \underline{c}\) and \(\underline{a}-3 \underline{b}-5 \underline{c}\) are coplanar. \(\mathrm{m}=\) (a) 2 (b) \(-1\) (c) 1 (d) \(-(9 / 7)\)

Short answer

\(\mathrm{m}=-\frac{9}{7}\) (d)

Step-by-step solution

Calculate the cross product of the second and third vectors

To find the cross product, we will use the formula \(\underline{u} \times \underline{v} = (u_2v_3-u_3v_2, u_3v_1-u_1v_3, u_1v_2-u_2v_1)\). So, we have: \[ [-2\underline{a}+3\underline{b}-4 \underline{c}] \times [\underline{a}-3 \underline{b}-5\underline{c}] = (A, B, C), \] where \[ A = (3\underline{b}-4\underline{c})\cdot(-5\underline{c}) - (-3\underline{b}-5\underline{c})\cdot(-4\underline{c}), \] \[ B = (-4\underline{c})\cdot\underline{a} - \underline{a}\cdot(-5\underline{c}), \] \[ C = \underline{a}\cdot(-3\underline{b}) - (-2\underline{a}+3\underline{b})\cdot(-3\underline{b}). \]

Calculate the dot product of the first vector and cross product from step 1

Now, we will evaluate the dot product of the first vector \((\underline{a}+m\underline{b}+3\underline{c})\) and the cross product \((A,B,C)\). The dot product is: \[ (\underline{a}+m \underline{b}+3 \underline{c}) \cdot (A, B, C) = \underline{a}\cdot A + m \underline{b} \cdot A + 3 \underline{c} \cdot A + \underline{a} \cdot B + m \underline{b} \cdot B + 3 \underline{c} \cdot B + \underline{a} \cdot C + m \underline{b} \cdot C + 3 \underline{c} \cdot C. \]

Solve the equation for m

As the scalar triple product is zero, we have: \[ \underline{a}\cdot A + m \underline{b} \cdot A + 3 \underline{c} \cdot A + \underline{a} \cdot B + m \underline{b} \cdot B + 3 \underline{c} \cdot B + \underline{a} \cdot C + m \underline{b} \cdot C + 3 \underline{c} \cdot C=0. \] Solving this equation for \(m\), we get: \[ m = -\frac{\underline{a}\cdot A + 3 \underline{c} \cdot A + \underline{a} \cdot B + 3 \underline{c} \cdot B + \underline{a} \cdot C + 3 \underline{c} \cdot C}{ \underline{b} \cdot A + \underline{b} \cdot B + \underline{b} \cdot C}. \] Plugging in the values of \(A\), \(B\), and \(C\) and calculating \(m\), we find that the value of m is \(-\frac{9}{7}\). Therefore, the solution is: \(\mathrm{m}=-\frac{9}{7}\) which corresponds to option (d).

Key concepts

Cross Product

The cross product, also known as the vector product, is a binary operation on two vectors in three-dimensional space. It provides a vector that is perpendicular to the plane in which the original vectors lie. This is particularly useful in physics and engineering for finding a normal to a surface or in determining torque.

The cross product of vectors \textbf{u} and \textbf{v}, denoted by \textbf{u} \times \textbf{v}, is given by the formula:
\[ \textbf{u} \times \textbf{v} = (u_2v_3-u_3v_2, u_3v_1-u_1v_3, u_1v_2-u_2v_1) \]
where, \textbf{u} and \textbf{v} have components \textbf{u} = (u_1, u_2, u_3) and \textbf{v} = (v_1, v_2, v_3). This formula yields a new vector \textbf{w} which stands orthogonal to both \textbf{u} and \textbf{v}. To comprehend the cross product, envision placing two fingers in the direction of the vectors; the resultant cross product points in the direction your thumb would extend, assuming a right-handed coordinate system. When determining the coplanarity of vectors, the cross product plays a crucial role since coplanar vectors will yield a cross product vector that is zero when dotted with any vector in their plane.

Dot Product

The dot product, or scalar product, is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation combines vectors in a way that reflects the extent to which they point in the same direction. It's a fundamental operation in the study of vectors and can be used to derive the angle between two vectors, project one vector onto another, or test for orthogonality.

The dot product of two vectors \textbf{u} and \textbf{v} is denoted by \textbf{u} \. \textbf{v} and is calculated as:
\[\textbf{u} \. \textbf{v} = u_1v_1 + u_2v_2 + u_3v_3\]
In a geometrical context, the dot product is deeply connected to the cosine of the angle \theta\ between the two vectors:
\[\textbf{u} \. \textbf{v} = |\textbf{u}| |\textbf{v}| \cos(\theta)\]
Where |\textbf{u}| and |\textbf{v}| are the magnitudes of \textbf{u} and \textbf{v}, respectively. If the dot product is zero, it indicates that the vectors are orthogonal to each other. In the context of checking for coplanarity, the dot product is used to establish a condition of perpendicularity between the cross product of two vectors and a third vector.

Scalar Triple Product

The scalar triple product is a measure of the volume of the parallelepiped formed by three vectors. It is calculated using the cross product and dot product of the vectors. The parallelepiped is a three-dimensional figure, and when the volume is zero, it implies that the vectors are coplanar, or lie in the same plane.

To find the scalar triple product of vectors \textbf{a}, \textbf{b}, and \textbf{c}, you calculate the dot product of one of the vectors with the cross product of the other two:
\[\textbf{a} \. (\textbf{b} \times \textbf{c})\]
This operation produces a scalar (hence the name 'scalar triple product'). If the result is zero, the vectors are coplanar. This property is essential in our exercise, as establishing the value for 'm' hinges upon the condition that the scalar triple product equals zero, indicating that the given vectors are indeed coplanar.

Vector Algebra

Vector algebra is the branch of mathematics that deals with vectors and the various operations that can be performed on them. These operations, including addition, subtraction, scalar multiplication, and products (dot and cross), allow us to solve complex problems in physics and engineering, such as those involving force, motion, and fields.

In vector algebra, the properties of vectors such as distributive, associative, and commutative laws are applied, similar to how they are in regular algebra, but with the consideration of both magnitude and direction. For instance:
- The sum of two vectors is the vector resulting from a 'head-to-tail' placing, visualized via the parallelogram rule.
- Multiplying a vector by a scalar changes its magnitude without altering its direction, unless the scalar is negative, in which case the direction is reversed.

In the context of our textbook exercise, vector algebra principles are employed to calculate the cross product and dot product, and to understand how these operations relate to the concept of coplanarity. Such practical applications of vector algebra are what make it a vital part of the curriculum in mathematical and physical sciences.