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Applications of Dot Product

Find the angl...

Question

Find the angle between the vectors $3 i + 2 j - 6 k$ and $4 i - 3 j + k$ .

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Solution

Consider the given vectors.

$a = 3 i + 2 j - 6 k$

$b = 4 i - 3 j + k$

Now,

$a \cdot b = (3 i + 2 j - 6 k) (4 i - 3 j + k)$

$a \cdot b = 12 - 6 - 6 = 0$

Therefore,

Two vectors $a$ and $b$ are perpendicular.

Thus, the angle between the vectors $a$ and $b$ $= 90^{\circ}$

Hence, this is the answer.

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Q. Find the angle between the vectors

\vec{a} and \vec{b}

\vec{a} = \hat{i} - \hat{j} and \vec{b} = \hat{j} + \hat{k}

\vec{a} = 3 \hat{i} - 2 \hat{j} - 6 \hat{k} and \vec{b} = 4 \hat{i} - \hat{j} + 8 \hat{k}

\vec{a} = 2 \hat{i} - \hat{j} + 2 \hat{k} and \vec{b} = 4 \hat{i} + 4 \hat{j} - 2 \hat{k}

\vec{a} = 2 \hat{i} - 3 \hat{j} + \hat{k} and \vec{b} = \hat{i} + \hat{j} - 2 \hat{k}

\vec{a} = \hat{i} + 2 \hat{j} - \hat{k}, \vec{b} = \hat{i} - \hat{j} + \hat{k}

Q. Find the angle between the vectors

ˆ i - 2 ˆ j + 3 ˆ k

3 ˆ i - 2 ˆ j + ˆ k

Q. Find the angle between the vectors

¯ a = 3 ˆ i - 2 ˆ j + ˆ k

¯ b = ˆ i - 2 ˆ j - 3 ˆ k

Q. Find the distance between the lines l₁ and l₂ given by

\vec{r} = \hat{i} + 2 \hat{j} - 4 \hat{k} + λ (2 \hat{i} + 3 \hat{j} + 6 \hat{k}) and, \vec{r} = 3 \hat{i} + 3 \hat{j} - 5 \hat{k} + μ (2 \hat{i} + 3 \hat{j} + 6 \hat{k})

\to a = 3 \to i - 2 \to j - 6 \to k

\to b = 4 \to i - \to j + 8 \to k

, then the angle between

2 \to a

3 \to b

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