Chapter 3: Q5P (page 136)

Show that any vector Vin a plane can be written as a linear combination of two non-parallel vectors Aand Bin the plane; that is, find aand bso that $V = aA + bB$ . Hint: Find the cross products $A \times V$ and $B \times V$ ; what are $A \times A$ and $B \times B$ ? Take components perpendicular to the plane to show that
$a = \frac{(B \times V) \cdot n}{(B \times A) \cdot n}$
Where,nisnormal to the plane, and a similar formula for b.

Short Answer

Expert verified

For finding a and b, take the cross product of A and B with V such that $a = \frac{(B \times v) \cdot n}{(B \times A) \cdot n}$ and $b = \frac{(A \times v) \cdot n}{(A \times B) \cdot n}$ .

Step by step solution

Definition of Cross Product of two vectors

In three-dimensional space, a cross product is a binary operation on two vectors. It yields a vector orthogonal to both vectors. $a \times b$ indicates the vector product of two vectors, aand b.

Given Parameters

The given vector equation is $V = aA + bB .$

Find a and b .

Finding the cross product

Find the cross product of A with V.

$\begin{array}{l} A \times V = A \times (aA + bB) \\ A \times V = bA \times 8 & (∵ A \times A = 0) \end{array}$

Find the cross product of B with V.

$\begin{array}{l} B \times v = B \times (aA + bB) \\ 8 \times V = aB \times A & (∵ B \times 8 = 0) \end{array}$

Finding the dot product

Find the dot product of $(A \times V)$ and n.

$(A \times V) \cdot n = b ((A \times B) \cdot n) b = \frac{(A \times V) \cdot n}{(A \times B) \cdot n}$

Find the dot product of $(B \times V)$ and n .

$(B \times V) \cdot n = a ((B \times A) \cdot n) a = \frac{(B \times V) \cdot n}{(B \times A) \cdot n}$

Therefore, any vector V in a plane can be written as a linear combination of two non-parallel vectors A and B in the plane; such that $V = aA + bB$ if $a = \frac{(B \times v) \cdot n}{(B \times A) \cdot n}$ and role="math" localid="1659073525714" $b = \frac{(A \times v) \cdot n}{(A \times B) \cdot n}$ ,

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Short Answer

Step by step solution

Definition of Cross Product of two vectors

Given Parameters

Finding the cross product

Finding the dot product

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