Chapter 3: Q36P (page 59)

If ${\overset{⇀}{d}}_{1} = 3 \hat{i} - 2 \hat{j} + 4 \hat{k}$ and ${\overset{⇀}{d}}_{2} = - 5 \hat{i} + 2 \hat{j} - \hat{k}$ then what is $({\overset{⇀}{d}}_{1} + {\overset{⇀}{d}}_{2}) . ({\overset{⇀}{d}}_{1} \times 4 {\overset{⇀}{d}}_{2})$ ?

Short Answer

Expert verified

The product, $({\overset{⇀}{d}}_{1} + {\overset{⇀}{d}}_{2}) . ({\overset{⇀}{d}}_{1} \times 4 {\overset{⇀}{d}}_{2})$ is equal to 0.

Step by step solution

Vector operations

The dot product of perpendicular vectors is equal to zero. The cross-product of two vectors results in a vector that is perpendicular to both vectors.

Find the value for $({\overset{⇀}{d}}_{1} + {\overset{⇀}{d}}_{2})$ and $({\overset{⇀}{d}}_{1} \times 4 {\overset{⇀}{d}}_{2})$ separately and then take the dot product of them to find the required answer.

The given vectors are,

$\overset{⇀}{d_{1}} = 3 \hat{i} - 2 \hat{j} + 4 \hat{k} {\overset{⇀}{d}}_{2} = - 5 \hat{j} + 2 \hat{j} - \hat{k}$

Calculating the cross and dot product

${\overset{⇀}{d}}_{1} + {\overset{⇀}{d}}_{2}$ will lie in the same plane i.e., in the plane of ${\overset{⇀}{d}}_{1} and {\overset{⇀}{d}}_{2}$ .

${\overset{⇀}{d}}_{1} \times 4 {\overset{⇀}{d}}_{2}$ will lie in the plane which is perpendicular to the plane of ${\overset{⇀}{d}}_{1} and {\overset{⇀}{d}}_{2}$ .

As they are perpendicular to each other the dot product of ${\overset{⇀}{d}}_{1} + {\overset{⇀}{d}}_{2}$ and ${\overset{⇀}{d}}_{1} \times 4 {\overset{⇀}{d}}_{2}$ will be zero.

role="math" localid="1656309525133" $({\overset{⇀}{d}}_{1} + {\overset{⇀}{d}}_{2})$ . $({\overset{⇀}{d}}_{1} \times 4 {\overset{⇀}{d}}_{2})$ =0

Thus, the dot product of $({\overset{⇀}{d}}_{1} + {\overset{⇀}{d}}_{2})$ and $({\overset{⇀}{d}}_{1} \times 4 {\overset{⇀}{d}}_{2})$ is equal to 0.

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If ${\overset{⇀}{d}}_{1} = 3 \hat{i} - 2 \hat{j} + 4 \hat{k}$ and ${\overset{⇀}{d}}_{2} = - 5 \hat{i} + 2 \hat{j} - \hat{k}$ then what is $({\overset{⇀}{d}}_{1} + {\overset{⇀}{d}}_{2}) . ({\overset{⇀}{d}}_{1} \times 4 {\overset{⇀}{d}}_{2})$ ?

Short Answer

Step by step solution

Vector operations

Calculating the cross and dot product

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If d⇀1=3i^-2j^+4k^and d⇀2=-5i^+2j^-k^then what is (d⇀1+d⇀2).(d⇀1×4d⇀2)?

Short Answer

Step by step solution

Vector operations

Calculating the cross and dot product

One App. One Place for Learning.

Most popular questions from this chapter

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If ${\overset{⇀}{d}}_{1} = 3 \hat{i} - 2 \hat{j} + 4 \hat{k}$ and ${\overset{⇀}{d}}_{2} = - 5 \hat{i} + 2 \hat{j} - \hat{k}$ then what is $({\overset{⇀}{d}}_{1} + {\overset{⇀}{d}}_{2}) . ({\overset{⇀}{d}}_{1} \times 4 {\overset{⇀}{d}}_{2})$ ?