Chapter 3: Q24P (page 106)

What is the value of $(A \times B)^{2} + (A \times B)^{2} ?$ Comment: This is a special case of Lagrange's identity. (See Chapter 6. Problem 3.12b, page 284.)
Use vectors as in Problems 3 to 8, and also the dot and cross product, to prove the following theorems from geometry.

Short Answer

Expert verified

The required value is $(\vec{A} \times \vec{B})^{2} + (\vec{A} \times \vec{B})^{2} = | \vec{A} |^{2} | \vec{B} |^{2} = A^{2} B^{2}$

Step by step solution

Concept and formula used

The vector product, and the scalar product of two vectors with $θ$ angle between them and unit vector $\hat{n}$ perpendicular to the plane containing them, is defined as:

Vector product: $\vec{A} \times \vec{B} = | \vec{A} | | \vec{B} | s i n θ \hat{n}$ .

Scalar product: $\vec{A} \times \vec{B} = | \vec{A} | | \vec{B} | c o s θ$ .

To find the value of the given vector

For two vectors $\vec{A}$ and $\vec{B}$ ,

$(\vec{A} \times \vec{B}) = | \vec{A} | | \vec{B} | s i n θ \hat{n} (\vec{A} \times \vec{B})^{2} = (| \vec{A} | | \vec{B} | s i n θ \hat{n})^{2} = | \vec{A} |^{2} | \vec{B} |^{2} s i n^{2} θ$

Therefore $(\vec{A} \times \vec{B})^{2} = A^{2} B^{2} s i n^{2} θ \dots (1)$

$Q | \vec{A} | = A, | \vec{B} | = B, (\hat{n})^{2} = \hat{n} \times \hat{n} = 1$

$(\vec{A} \times \vec{B}) = | \vec{A} | | \vec{B} | c o s θ (\vec{A} \times \vec{B})^{2} = (| \vec{A} | | \vec{B} | c o s θ \hat{n})^{2} = | \vec{A} |^{2} | \vec{B} |^{2} c o s^{2} θ (\vec{A} \times \vec{B})^{2} = A^{2} B^{2} c o s^{2} θ \dots (2)$

From results (1) and (2)

$(\vec{A} \times \vec{B})^{2} + (\vec{A} \times \vec{B})^{2} = | \vec{A} |^{2} | \vec{B} |^{2} s i n^{2} θ + | \vec{A} |^{2} | \vec{B} |^{2} c o s^{2} θ = | \vec{A} |^{2} | \vec{B} |^{2} = A^{2} B^{2}$

Thus $(\vec{A} \times \vec{B})^{2} + (\vec{A} \times \vec{B})^{2} = | \vec{A} |^{2} | \vec{B} |^{2} = A^{2} B^{2}$

Hence, the required value is $(\vec{A} \times \vec{B})^{2} + (\vec{A} \times \vec{B})^{2} = | \vec{A} |^{2} | \vec{B} |^{2} = A^{2} B^{2}$ .

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What is the value of $(A \times B)^{2} + (A \times B)^{2} ?$ Comment: This is a special case of Lagrange's identity. (See Chapter 6. Problem 3.12b, page 284.)
Use vectors as in Problems 3 to 8, and also the dot and cross product, to prove the following theorems from geometry.

Short Answer

Step by step solution

Concept and formula used

To find the value of the given vector

One App. One Place for Learning.

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What is the value of (A×B)2+(A×B)2?Comment: This is a special case of Lagrange's identity. (See Chapter 6. Problem 3.12b, page 284.)Use vectors as in Problems 3 to 8, and also the dot and cross product, to prove the following theorems from geometry.

Short Answer

Step by step solution

Concept and formula used

To find the value of the given vector

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Physical Quantities And Units

Medical Physics

Fluids

Dynamics

Astrophysics

Particle Model of Matter

Study anywhere. Anytime. Across all devices.

What is the value of $(A \times B)^{2} + (A \times B)^{2} ?$ Comment: This is a special case of Lagrange's identity. (See Chapter 6. Problem 3.12b, page 284.)
Use vectors as in Problems 3 to 8, and also the dot and cross product, to prove the following theorems from geometry.