The problem involves vector algebra and geometry. It includes finding magnitudes of vectors, dot products, direction vectors, and scalar components. The given vectors are $V = 2i + 4j + \sqrt{5}k$ and $U = -2i + 4j - \sqrt{5}k$. We need to find the cosine of the angle between the two vectors and the scalar component of $U$ in the direction of $V$.

1. Problem Description

The problem involves vector algebra and geometry. It includes finding magnitudes of vectors, dot products, direction vectors, and scalar components. The given vectors are $V = 2i + 4j + \sqrt{5}k$ and $U = -2i + 4j - \sqrt{5}k$. We need to find the cosine of the angle between the two vectors and the scalar component of $U$ in the direction of $V$.

2. Solution Steps

(i) We are given $V = 2i + 4j + \sqrt{5}k$ and $U = -2i + 4j - \sqrt{5}k$. We have already found that $|V| = \sqrt{2^2 + 4^2 + (\sqrt{5})^2} = \sqrt{4 + 16 + 5} = \sqrt{25} = 5$ and $|U| = \sqrt{(-2)^2 + 4^2 + (-\sqrt{5})^2} = \sqrt{4 + 16 + 5} = \sqrt{25} = 5$.

Also, $V \cdot U = (2)(-2) + (4)(4) + (\sqrt{5})(-\sqrt{5}) = -4 + 16 - 5 = 7$. There appears to be an error in the image's calculation of V.U. It states V.U = -25, but the correct calculation is

7.

(ii) The cosine of the angle between $V$ and $U$ is given by the formula:

$cos(\theta) = \frac{V \cdot U}{|V||U|}$

Plugging in the values, we get:

$cos(\theta) = \frac{7}{5 \cdot 5} = \frac{7}{25}$

The solution in the image states the $cos(\theta) = -1$ which is incorrect.

(iii) The scalar component of $U$ in the direction of $V$ is given by the formula:

$comp_V U = \frac{U \cdot V}{|V|}$

Plugging in the values, we get:

$comp_V U = \frac{7}{5}$

The scalar component calculated in the image is also incorrect based on the wrong calculation of $V.U$. The correct answer should be $7/5$, not -

5.

3. Final Answer

(i) $|V| = 5$, $|U| = 5$, $V \cdot U = 7$

(ii) $cos(\theta) = \frac{7}{25}$

(iii) Scalar component of $U$ in the direction of $V$: $\frac{7}{5}$