Question

Two adjacent sides of a parallelogram \(\mathrm{ABCD}\) are given by \(\overrightarrow{\mathrm{AB}}=2 \hat{\mathrm{i}}+10 \hat{\mathrm{j}}+11 \hat{\mathrm{k}}\) and \(\overrightarrow{\mathrm{AD}}=-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}\) The side \(\mathrm{AD}\) is rotated by an acute angle \(\alpha\) in the plane of the parallelogram so that \(\mathrm{AD}\) becomes \(\mathrm{AD}^{\prime}\). If \(\mathrm{AD}^{\prime}\) makes a right angle with the side \(\mathrm{AB}\), then the cosine of the angle \(\alpha\) is given by A) \(\frac{8}{9}\) B) \(\frac{\sqrt{17}}{9}\) C) \(\frac{1}{9}\) D) \(\frac{4 \sqrt{5}}{9}\)

Short answer

\(\frac{8}{9}\)

Step-by-step solution

Determine Dot Product of Original Vectors

Calculate the dot product of the vectors \(\overrightarrow{\mathrm{AB}}\) and \(\overrightarrow{\mathrm{AD}}\) to find out the cosine of the angle between \(\mathrm{AB}\) and \(\mathrm{AD}\) before rotation. The formula for the dot product is \(\overrightarrow{\mathrm{A}} \cdot \overrightarrow{\mathrm{B}} = A_xB_x + A_yB_y + A_zB_z\) where subscripts denote the components along \(\hat{\mathrm{i}}\), \(\hat{\mathrm{j}}\), and \(\hat{\mathrm{k}}\) respectively.

Calculate Original Cosine Value

Use the dot product found in Step 1 to calculate the cosine of the angle between \(\overrightarrow{\mathrm{AB}}\) and \(\overrightarrow{\mathrm{AD}}\) initially using the formula \(\cos(\theta) = \frac{\overrightarrow{\mathrm{A}} \cdot \overrightarrow{\mathrm{B}}}{|\overrightarrow{\mathrm{A}}||\overrightarrow{\mathrm{B}}|}\) where |\overrightarrow{\mathrm{A}}| and |\overrightarrow{\mathrm{B}}| are the magnitudes of the vectors.

Determine the Magnitude of Vectors

Find the magnitudes of vectors \(\overrightarrow{\mathrm{AB}}\) and \(\overrightarrow{\mathrm{AD}}\) using the formula \(\|\overrightarrow{A}\| = \sqrt{A_x^2 + A_y^2 + A_z^2}\).

Find Cosine of Rotation Angle

Since \(\mathrm{AD}'\) is perpendicular to \(\mathrm{AB}\), the dot product of \(\overrightarrow{\mathrm{AD}'}\) and \(\overrightarrow{\mathrm{AB}}\) must be 0. Knowing that \(\overrightarrow{\mathrm{AD}}\) and \(\overrightarrow{\mathrm{AD}'}\) form an angle \(\alpha\), use the relationship that the cosine of the angle between two vectors is also the cosine of the angle of rotation to find the cosine of \(\alpha\).

Compare with Given Options

After finding the value of \(\cos(\alpha)\) in the previous step, compare it with the options given to find the correct answer.

Key concepts

Dot Product of Vectors

The dot product, also known as the scalar product, is a fundamental operation in vector algebra, particularly useful when dealing with parallelograms like in our example involving vectors \(\overrightarrow{\mathrm{AB}}\) and \(\overrightarrow{\mathrm{AD}}\). It's defined for two vectors \(\overrightarrow{\mathrm{A}}\) and \(\overrightarrow{\mathrm{B}}\) as \(\overrightarrow{\mathrm{A}} \cdot \overrightarrow{\mathrm{B}} = A_xB_x + A_yB_y + A_zB_z\), where the subscripts denote their respective components.

When we calculate the dot product, it gives us a scalar quantity that represents the product of the vectors' magnitudes and the cosine of the angle between them. This property is especially useful because we can use it to determine if two vectors are perpendicular (if their dot product is zero) or to find the angle between them. It's a critical step for understanding how the vectors relate to each other in the context of our parallelogram exercise.

Vector Rotation Angle

Understanding vector rotation is essential when dealing with parallelograms in vector algebra. When a vector like \(\overrightarrow{\mathrm{AD}}\) is rotated to become \(\overrightarrow{\mathrm{AD}'}\), it forms a new angle \(\alpha\) with its original position. The concept of vector rotation angle allows us to determine the orientation of a vector after it has been rotated within a plane.

If the rotated vector \(\overrightarrow{\mathrm{AD}'}\) becomes orthogonal to another vector \(\overrightarrow{\mathrm{AB}}\), as in our exercise, we can infer that the dot product between these two vectors is zero. More importantly, finding the cosine of the rotation angle \(\alpha\) tells us the extent of this rotation. This concept is critical when we need to solve problems involving relative positions and angles within geometric figures such as parallelograms.

Magnitude of Vectors

The magnitude of a vector signifies its length or size and is another fundamental element of vector algebra. In our parallelogram-related exercise, the magnitude is used to calculate the scale of the sides represented by vectors \(\overrightarrow{\mathrm{AB}}\) and \(\overrightarrow{\mathrm{AD}}\).

The formula to compute a vector's magnitude is \(\|\overrightarrow{A}\| = \sqrt{A_x^2 + A_y^2 + A_z^2}\), which derives from the Pythagorean theorem. This results in a scalar expressing how far the vector extends in the multidimensional space it occupies. When dealing with vectors like sides of a parallelogram, knowing their magnitudes helps us understand the proportions and dimensions of the shape.

Cosine of Angle Between Vectors

The cosine of the angle between two vectors is a measure that indicates the directional similarity of the vectors. Calculating this value plays a key role in our understanding of the relationship between the sides of a parallelogram represented by the vectors.

In the context of our exercise, we use the formula \(\cos(\theta) = \frac{\overrightarrow{\mathrm{A}} \cdot \overrightarrow{\mathrm{B}}}{|\overrightarrow{\mathrm{A}}||\overrightarrow{\mathrm{B}}|}\), where \(\theta\) is the angle between vectors \(\overrightarrow{\mathrm{A}}\) and \(\overrightarrow{\mathrm{B}}\), and the bars represent the magnitudes of the vectors. When vectors are perpendicular, as stipulated in the case of our rotated vector \(\overrightarrow{\mathrm{AD}'}\) making a right angle with \(\overrightarrow{\mathrm{AB}}\), the cosine of the angle between them will be zero, reinforcing the importance of this concept to analyze geometrical configurations in vector spaces.