The length of the shadow for \( v \) in the figure shown is the absolute value of the vector's horizontal component. The magnitude and direction angle of \( v \) are given. Write \( v \) in terms of \( i \) and \( j \). Then find the length of the shadow. \[ v \|=1.4 \text { inches, } \theta=32^{\circ} \] \( v=\square \) (Type your answer in terms of \( i \) and \( j \). Round to the nearest tenth as needed.) The length of the shadow is \( \square \) inches. (Round to the nearest tenth as needed.)

To express vector \( v \) in terms of \( i \) and \( j \), we can use the magnitude and direction angle. The horizontal component can be calculated using the cosine of the angle, and the vertical component using the sine: \[ v = v \cos(\theta) i + v \sin(\theta) j \] Substituting the values: \[ v = 1.4 \cos(32^{\circ}) i + 1.4 \sin(32^{\circ}) j \] Calculating the components: - For the horizontal component (using cosine): \[ 1.4 \cos(32^{\circ}) \approx 1.4 \times 0.846 = 1.1835 \quad (\text{rounding to } 1.2) \] - For the vertical component (using sine): \[ 1.4 \sin(32^{\circ}) \approx 1.4 \times 0.529 = 0.7406 \quad (\text{rounding to } 0.7) \] Therefore, the vector can be written as: \[ v \approx 1.2 i + 0.7 j \] The length of the shadow is equal to the absolute value of the horizontal component, which we calculated as approximately \( 1.2 \) inches. So, the answers are: \( v \approx 1.2 i + 0.7 j \) The length of the shadow is approximately \( 1.2 \) inches.